User:Doug Bell/Probability derivations for making low hands in Omaha hold 'em


 * See Poker probability (Omaha) for the probabilities derived from these equations.

The probability derivations for starting hands making a qualifying low hand in Omaha hold 'em Hi-Lo are separate for each of the starting hand low shapes. The derivations require identifying the individual cases that complete a low hand; make a low hand possible on the board, but don't make a low hand; and that create a board with fewer than three low ranks, making no low hand possible.

The ranks on the board are indicated using lower case letters for matches with the starting hand and upper case letters to indicate qualifying low ranks that don't match the starting hand. So the low hand shape xxyz is any hand with a low pair of x with two additional low ranks y and z and the board xyA represents a flop that contains one x, one of the non-paired ranks y and one other low rank A. Note that since y and z have an identical relationship to the starting hand—each represents an unpaired rank—xyA and xzA represent the same set of boards and are interchangeable, so derivations for this hand choose one of the two choices represented by y. In addition to the upper and lower case letters, * is used to represent any rank higher than the qualifying low. So for the low hand shape xxy*, the board xx* represents a flop that contains two xs and a high rank (either the same high rank as xxy* or a different high rank).

One common calculation regardless of the hand is that no low is possible when there are two or fewer low cards on the board. If $$h$$ is the number of high cards in the hand ($$h$$ = 0, 1, 2, 3, 4) and $$r$$ is the low qualifier ($$r$$ = 8, 9), then $$4r - 4 + h$$ is the number of low cards remaining in the deck and $$52 - 4r - h$$ is the number of high cards remaining in the deck. Using ? to represent any low card, the number of boards with 0, 1 or 2 low cards on the flop, turn and river are calculated as follows:


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! colspan=2| Flop !! colspan=2| Turn !! colspan=2| River ! Board !! Derivation !! Board !! Derivation !! Board !! Derivation
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {4r-4+h \choose 2}{52-4r-h \choose 1} \end{matrix}$$
 * $$\begin{matrix} {4r-4+h \choose 2}{52-4r-h \choose 2} \end{matrix}$$
 * $$\begin{matrix} {4r-4+h \choose 2}{52-4r-h \choose 3} \end{matrix}$$
 * $$\begin{matrix} {4r-4+h \choose 1}{52-4r-h \choose 2} \end{matrix}$$
 * $$\begin{matrix} {4r-4+h \choose 1}{52-4r-h \choose 3} \end{matrix}$$
 * $$\begin{matrix} {4r-4+h \choose 1}{52-4r-h \choose 4} \end{matrix}$$
 * $$\begin{matrix} {52-4r-h \choose 3} \end{matrix}$$
 * $$\begin{matrix} {52-4r-h \choose 4} \end{matrix}$$
 * $$\begin{matrix} {52-4r-h \choose 5} \end{matrix}$$
 * }
 * $$\begin{matrix} {4r-4+h \choose 1}{52-4r-h \choose 3} \end{matrix}$$
 * $$\begin{matrix} {4r-4+h \choose 1}{52-4r-h \choose 4} \end{matrix}$$
 * $$\begin{matrix} {52-4r-h \choose 3} \end{matrix}$$
 * $$\begin{matrix} {52-4r-h \choose 4} \end{matrix}$$
 * $$\begin{matrix} {52-4r-h \choose 5} \end{matrix}$$
 * }
 * $$\begin{matrix} {52-4r-h \choose 3} \end{matrix}$$
 * $$\begin{matrix} {52-4r-h \choose 4} \end{matrix}$$
 * $$\begin{matrix} {52-4r-h \choose 5} \end{matrix}$$
 * }
 * $$\begin{matrix} {52-4r-h \choose 5} \end{matrix}$$
 * }

Each table in the following subsections shows all of the boards that can make each hand and the derivation for the combinations for that board. Probabilities are determined by dividing the number of combinations for each hand by the $$\begin{matrix} {48 \choose 3} = 17,296 \end{matrix}$$ boards on the flop, $$\begin{matrix} {48 \choose 4} = 194,580 \end{matrix}$$ boards on the turn, and $$\begin{matrix} {48 \choose 5} = 1,712,304 \end{matrix}$$ boards at the river. The probabilities for the boards in each table total 1.0.

Derivations for making low hands with four high cards
Starting hands with four high cards (****) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With four high cards in the hand, there are $$52 - 4r - 4 = 48 - 4r$$ high cards and $$4r - 4 + 4 = 4r$$ low cards left in the deck. The following tables show the derivations when holding four high cards.


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=1| Miss low ! ABC ! rowspan=6| No low ! AAA ! AAB ! ??* ! ?** ! ***
 * +Derivations for low hand shape **** (four high cards) on the flop
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   3,584 || 0.2072155 ||align=right|      3.8 : 1
 * align=right|   5,376 || 0.3108233 ||align=right|      2.2 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      32 || 0.0018501 ||align=right|    539.5 : 1
 * align=right|      36 || 0.0020814 ||align=right|    479.4 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,344 || 0.0777058 ||align=right|     11.9 : 1
 * align=right|   1,728 || 0.0999075 ||align=right|      9.0 : 1
 * $$\begin{matrix} {4r \choose 2}{48-4r \choose 1} \end{matrix}$$
 * align=right|   7,168 || 0.4144311 ||align=right|      1.4 : 1
 * align=right|   6,912 || 0.3996300 ||align=right|      1.5 : 1
 * $$\begin{matrix} {4r \choose 1}{48-4r \choose 2} \end{matrix}$$
 * align=right|   7,936 || 0.4588344 ||align=right|      1.2 : 1
 * align=right|   7,560 || 0.4370953 ||align=right|      1.3 : 1
 * $$\begin{matrix} {48-4r \choose 3} \end{matrix}$$
 * align=right|     560 || 0.0323774 ||align=right|     29.9 : 1
 * align=right|     220 || 0.0127197 ||align=right|     77.6 : 1
 * colspan=2| Total
 * align=right|  13,712 || 0.7927845 ||align=right|      0.3 : 1
 * align=right|  11,920 || 0.6891767 ||align=right|      0.5 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=4| Miss low ! ABCD ! AABC ! ABC* ! rowspan=9| No low ! AAAA ! AAAB ! AABB ! AAA* ! AAB* ! ??** ! ?*** ! ****
 * +Derivations for low hand shape **** (four high cards) on the turn
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|  17,920 || 0.0920958 ||align=right|      9.9 : 1
 * align=right|  32,256 || 0.1657724 ||align=right|      5.0 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 2}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  16,128 || 0.0828862 ||align=right|     11.1 : 1
 * align=right|  24,192 || 0.1243293 ||align=right|      7.0 : 1
 * $$\begin{matrix} {r \choose 3}{4 \choose 1}^3{48-4r \choose 1} \end{matrix}$$
 * align=right|  57,344 || 0.2947065 ||align=right|      2.4 : 1
 * align=right|  64,512 || 0.3315449 ||align=right|      2.0 : 1
 * colspan=2| Total
 * align=right|  91,392 || 0.7927845 ||align=right|      1.1 : 1
 * align=right| 120,960 || 0.6216466 ||align=right|      0.6 : 1
 * align=right| 120,960 || 0.6216466 ||align=right|      0.6 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       8 || 0.0000411 ||align=right| 24,321.5 : 1
 * align=right|       9 || 0.0000463 ||align=right| 21,619.0 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     896 || 0.0046048 ||align=right|    216.2 : 1
 * align=right|   1,152 || 0.0059204 ||align=right|    167.9 : 1
 * $$\begin{matrix} {r \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|   1,008 || 0.0051804 ||align=right|    192.0 : 1
 * align=right|   1,296 || 0.0066605 ||align=right|    149.1 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 3}{48-4r \choose 1} \end{matrix}$$
 * align=right|     512 || 0.0026313 ||align=right|    379.0 : 1
 * align=right|     432 || 0.0022202 ||align=right|    449.4 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 1} \end{matrix}$$
 * align=right|  21,504 || 0.1105150 ||align=right|      8.0 : 1
 * align=right|  20,736 || 0.1065680 ||align=right|      8.4 : 1
 * $$\begin{matrix} {4r \choose 2}{48-4r \choose 2} \end{matrix}$$
 * align=right|  59,520 || 0.3058896 ||align=right|      2.3 : 1
 * align=right|  41,580 || 0.2136910 ||align=right|      3.7 : 1
 * $$\begin{matrix} {4r \choose 1}{48-4r \choose 3} \end{matrix}$$
 * align=right|  17,920 || 0.0920958 ||align=right|      9.9 : 1
 * align=right|   7,920 || 0.0407031 ||align=right|     23.6 : 1
 * $$\begin{matrix} {48-4r \choose 4} \end{matrix}$$
 * align=right|   1,820 || 0.0093535 ||align=right|    105.9 : 1
 * align=right|     495 || 0.0025439 ||align=right|    392.1 : 1
 * colspan=2| Total
 * align=right| 103,188 || 0.5303114 ||align=right|      0.9 : 1
 * align=right|  73,620 || 0.3783534 ||align=right|      1.6 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=8| Miss low ! ABCDE ! AABCD ! ABCD* ! AABBC ! AAABC ! AABC* ! ABC** ! rowspan=11| No low ! AAAAB ! AAAA* ! AAABB ! AAAB* ! AABB* ! AAA** ! AAB** ! ??*** ! ?**** ! *****
 * +Derivations for low hand shape **** (four high cards) on the river
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r \choose 5}{4 \choose 1}^5 \end{matrix}$$
 * align=right|  57,344 || 0.0334894 ||align=right|     28.9 : 1
 * align=right| 129,024 || 0.0753511 ||align=right|     12.3 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 2}{r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right| 107,520 || 0.0627926 ||align=right|     14.9 : 1
 * align=right| 193,536 || 0.1130267 ||align=right|      7.8 : 1
 * $$\begin{matrix} {r \choose 4}{4 \choose 1}^4{48-4r \choose 1} \end{matrix}$$
 * align=right| 286,720 || 0.1674469 ||align=right|      5.0 : 1
 * align=right| 387,072 || 0.2260533 ||align=right|      3.4 : 1
 * $$\begin{matrix} {r \choose 2}{4 \choose 2}^2{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|  24,192 || 0.0141283 ||align=right|     69.8 : 1
 * align=right|  36,288 || 0.0211925 ||align=right|     46.2 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 3}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  10,752 || 0.0062793 ||align=right|    158.3 : 1
 * align=right|  16,128 || 0.0094189 ||align=right|    105.2 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 2}{r-1 \choose 2}{4 \choose 1}^2{48-4r \choose 1} \end{matrix}$$
 * align=right| 258,048 || 0.1507022 ||align=right|      5.6 : 1
 * align=right| 290,304 || 0.1695400 ||align=right|      4.9 : 1
 * $$\begin{matrix} {r \choose 3}{4 \choose 1}^3{48-4r \choose 2} \end{matrix}$$
 * align=right| 430,080 || 0.2511704 ||align=right|      3.0 : 1
 * align=right| 354,816 || 0.2072155 ||align=right|      3.8 : 1
 * colspan=2| Total
 * align=right|1,174,656 || 0.6860090 ||align=right|     0.5 : 1
 * align=right|1,407,168 || 0.8217980 ||align=right|     0.2 : 1
 * align=right|1,407,168 || 0.8217980 ||align=right|     0.2 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 4}{r-1 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     224 || 0.0001308 ||align=right|  7,643.2 : 1
 * align=right|     288 || 0.0001682 ||align=right|  5,944.5 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 4}{48-4r \choose 1} \end{matrix}$$
 * align=right|     128 || 0.0000748 ||align=right| 13,376.4 : 1
 * align=right|     108 || 0.0000631 ||align=right| 15,853.7 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|   1,344 || 0.0007849 ||align=right|  1,273.0 : 1
 * align=right|   1,728 || 0.0010092 ||align=right|    989.9 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 3}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 1} \end{matrix}$$
 * align=right|  14,336 || 0.0083723 ||align=right|    118.4 : 1
 * align=right|  13,824 || 0.0080733 ||align=right|    122.9 : 1
 * $$\begin{matrix} {r \choose 2}{4 \choose 2}^2{48-4r \choose 1} \end{matrix}$$
 * align=right|  16,128 || 0.0094189 ||align=right|    105.2 : 1
 * align=right|  15,552 || 0.0090825 ||align=right|    109.1 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 3}{48-4r \choose 2} \end{matrix}$$
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * align=right|   2,376 || 0.0013876 ||align=right|    719.7 : 1
 * $$\begin{matrix} {r \choose 1}{4 \choose 2}{r-1 \choose 1}{4 \choose 1}{48-4r \choose 2} \end{matrix}$$
 * align=right| 161,280 || 0.0941889 ||align=right|      9.6 : 1
 * align=right| 114,048 || 0.0666050 ||align=right|     14.0 : 1
 * $$\begin{matrix} {4r \choose 2}{48-4r \choose 3} \end{matrix}$$
 * align=right| 277,760 || 0.1622142 ||align=right|      5.2 : 1
 * align=right| 138,600 || 0.0809436 ||align=right|     11.4 : 1
 * $$\begin{matrix} {4r \choose 1}{48-4r \choose 4} \end{matrix}$$
 * align=right|  58,240 || 0.0340127 ||align=right|     28.4 : 1
 * align=right|  17,820 || 0.0104070 ||align=right|     95.1 : 1
 * $$\begin{matrix} {48-4r \choose 5} \end{matrix}$$
 * align=right|   4,368 || 0.0025509 ||align=right|    391.0 : 1
 * align=right|     792 || 0.0004625 ||align=right|  2,161.0 : 1
 * colspan=2| Total
 * align=right| 537,648 || 0.3139910 ||align=right|      2.2 : 1
 * align=right| 305,136 || 0.1782020 ||align=right|      4.6 : 1
 * }
 * }

Derivations for making low hands with one low card and three high cards
Starting hands with one low card and three high cards (x***) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With three high cards in the hand, there are $$52 - 4r - 3 = 49 - 4r$$ high cards and $$4r - 4 + 3 = 4r - 1$$ low cards left in the deck. The following tables show the derivations when holding one low card and three high cards.


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=3| Miss low ! ABC ! xAB ! rowspan=9| No low ! AAA ! AAB ! xxx ! xxA ! xAA ! ??* ! ?** ! ***
 * +Derivations for low hand shape x*** (one low card and three high cards) on the flop
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   2,240 || 0.1295097 ||align=right|      6.7 : 1
 * align=right|   3,584 || 0.2072155 ||align=right|      3.8 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,008 || 0.0582794 ||align=right|     16.2 : 1
 * align=right|   1,344 || 0.0777058 ||align=right|     11.9 : 1
 * colspan=2| Total
 * align=right|   3,248 || 0.1877891 ||align=right|      4.3 : 1
 * align=right|   4,928 || 0.2849214 ||align=right|      2.5 : 1
 * align=right|   4,928 || 0.2849214 ||align=right|      2.5 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      28 || 0.0016189 ||align=right|    616.7 : 1
 * align=right|      32 || 0.0018501 ||align=right|    539.5 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,008 || 0.0582794 ||align=right|     16.2 : 1
 * align=right|   1,344 || 0.0777058 ||align=right|     11.9 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3} \end{matrix}$$
 * align=right|       1 || 0.0000578 ||align=right| 17,295.0 : 1
 * align=right|       1 || 0.0000578 ||align=right| 17,295.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      84 || 0.0048566 ||align=right|    204.9 : 1
 * align=right|      96 || 0.0055504 ||align=right|    179.2 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     126 || 0.0072849 ||align=right|    136.3 : 1
 * align=right|     144 || 0.0083256 ||align=right|    119.1 : 1
 * $$\begin{matrix} {4r-1 \choose 2}{49-4r \choose 1} \end{matrix}$$
 * align=right|   7,905 || 0.4570421 ||align=right|      1.2 : 1
 * align=right|   7,735 || 0.4472132 ||align=right|      1.2 : 1
 * $$\begin{matrix} {4r-1 \choose 1}{49-4r \choose 2} \end{matrix}$$
 * align=right|   4,216 || 0.2437558 ||align=right|      3.1 : 1
 * align=right|   2,730 || 0.1578400 ||align=right|      5.3 : 1
 * $$\begin{matrix} {49-4r \choose 3} \end{matrix}$$
 * align=right|     680 || 0.0393154 ||align=right|     24.4 : 1
 * align=right|     286 || 0.0165356 ||align=right|     59.5 : 1
 * colspan=2| Total
 * align=right|  14,048 || 0.8122109 ||align=right|      0.2 : 1
 * align=right|  12,368 || 0.7150786 ||align=right|      0.4 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=8| Miss low ! ABCD ! AABC ! ABC* ! xxAB ! xABC ! xAAB ! xAB* ! rowspan=15| No low ! AAAA ! AAAB ! AABB ! AAA* ! AAB* ! xxx* ! xxxA ! xxAA ! xxA* ! xAAA ! xAA* ! ??** ! ?*** ! ****
 * +Derivations for low hand shape x*** (one low card and three high cards) on the turn
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   8,960 || 0.0460479 ||align=right|     20.7 : 1
 * align=right|  17,920 || 0.0920958 ||align=right|      9.9 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  10,080 || 0.0518039 ||align=right|     18.3 : 1
 * align=right|  16,128 || 0.0828862 ||align=right|     11.1 : 1
 * $$\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{49-4r \choose 1} \end{matrix}$$
 * align=right|  38,080 || 0.1957036 ||align=right|      4.1 : 1
 * align=right|  46,592 || 0.2394491 ||align=right|      3.2 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,008 || 0.0051804 ||align=right|    192.0 : 1
 * align=right|   1,344 || 0.0069072 ||align=right|    143.8 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   6,720 || 0.0345359 ||align=right|     28.0 : 1
 * align=right|  10,752 || 0.0552575 ||align=right|     17.1 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   3,024 || 0.0155412 ||align=right|     63.3 : 1
 * align=right|   4,032 || 0.0207216 ||align=right|     47.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{49-4r \choose 1} \end{matrix}$$
 * align=right|  17,136 || 0.0880666 ||align=right|     10.4 : 1
 * align=right|  17,472 || 0.0897934 ||align=right|     10.1 : 1
 * colspan=2| Total
 * align=right|  85,008 || 0.4368794 ||align=right|      1.3 : 1
 * align=right| 114,240 || 0.5871107 ||align=right|      0.7 : 1
 * align=right| 114,240 || 0.5871107 ||align=right|      0.7 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       7 || 0.0000360 ||align=right| 27,796.1 : 1
 * align=right|       8 || 0.0000411 ||align=right| 24,321.5 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     672 || 0.0034536 ||align=right|    288.6 : 1
 * align=right|     896 || 0.0046048 ||align=right|    216.2 : 1
 * $$\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     756 || 0.0038853 ||align=right|    256.4 : 1
 * align=right|   1,008 || 0.0051804 ||align=right|    192.0 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{49-4r \choose 1} \end{matrix}$$
 * align=right|     476 || 0.0024463 ||align=right|    407.8 : 1
 * align=right|     416 || 0.0021379 ||align=right|    466.7 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1} \end{matrix}$$
 * align=right|  17,136 || 0.0880666 ||align=right|     10.4 : 1
 * align=right|  17,472 || 0.0897934 ||align=right|     10.1 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{49-4r \choose 1} \end{matrix}$$
 * align=right|      17 || 0.0000874 ||align=right| 11,444.9 : 1
 * align=right|      13 || 0.0000668 ||align=right| 14,966.7 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      28 || 0.0001439 ||align=right|  6,948.3 : 1
 * align=right|      32 || 0.0001645 ||align=right|  6,079.6 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     126 || 0.0006475 ||align=right|  1,543.3 : 1
 * align=right|     144 || 0.0007401 ||align=right|  1,350.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 1} \end{matrix}$$
 * align=right|   1,428 || 0.0073389 ||align=right|    135.3 : 1
 * align=right|   1,248 || 0.0064138 ||align=right|    154.9 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      84 || 0.0004317 ||align=right|  2,315.4 : 1
 * align=right|      96 || 0.0004934 ||align=right|  2,025.9 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 1} \end{matrix}$$
 * align=right|   2,142 || 0.0110083 ||align=right|     89.8 : 1
 * align=right|   1,872 || 0.0096207 ||align=right|    102.9 : 1
 * $$\begin{matrix} {4r-1 \choose 2}{49-4r \choose 2} \end{matrix}$$
 * align=right|  63,240 || 0.3250077 ||align=right|      2.1 : 1
 * align=right|  46,410 || 0.2385137 ||align=right|      3.2 : 1
 * $$\begin{matrix} {4r-1 \choose 1}{49-4r \choose 3} \end{matrix}$$
 * align=right|  21,080 || 0.1083359 ||align=right|      8.2 : 1
 * align=right|  10,010 || 0.0514441 ||align=right|     18.4 : 1
 * $$\begin{matrix} {49-4r \choose 4} \end{matrix}$$
 * align=right|   2,380 || 0.0122315 ||align=right|     80.8 : 1
 * align=right|     715 || 0.0036746 ||align=right|    271.1 : 1
 * colspan=2| Total
 * align=right| 109,572 || 0.5631206 ||align=right|      0.8 : 1
 * align=right|  80,340 || 0.4128893 ||align=right|      1.4 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=19| Miss low ! ABCDE ! AABCD ! ABCD* ! AABBC ! AAABC ! AABC* ! ABC** ! xABCD ! xABC* ! xAABC ! xAAB* ! xAAAB ! xAABB ! xAB** ! xxABC ! xxAB* ! xxAAB ! xxxAB ! rowspan=20| No low ! AAAAB ! AAAA* ! AAABB ! AAAB* ! AABB* ! AAA** ! AAB** ! xxxAA ! xxxA* ! xxx** ! xxAAA ! xxAA* ! xxA** ! xAAAA ! xAAA* ! xAA** ! ??*** ! ?**** ! *****
 * +Derivations for low hand shape x*** (one low card and three high cards) on the river
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-1 \choose 5}{4 \choose 1}^5 \end{matrix}$$
 * align=right|  21,504 || 0.0125585 ||align=right|     78.6 : 1
 * align=right|  57,344 || 0.0334894 ||align=right|     28.9 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|  53,760 || 0.0313963 ||align=right|     30.9 : 1
 * align=right| 107,520 || 0.0627926 ||align=right|     14.9 : 1
 * $$\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4{49-4r \choose 1} \end{matrix}$$
 * align=right| 152,320 || 0.0889562 ||align=right|     10.2 : 1
 * align=right| 232,960 || 0.1360506 ||align=right|      6.4 : 1
 * $$\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|  15,120 || 0.0088302 ||align=right|    112.2 : 1
 * align=right|  24,192 || 0.0141283 ||align=right|     69.8 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   6,720 || 0.0039245 ||align=right|    253.8 : 1
 * align=right|  10,752 || 0.0062793 ||align=right|    158.3 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{49-4r \choose 1} \end{matrix}$$
 * align=right| 171,360 || 0.1000757 ||align=right|      9.0 : 1
 * align=right| 209,664 || 0.1224455 ||align=right|      7.2 : 1
 * $$\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{49-4r \choose 2} \end{matrix}$$
 * align=right| 304,640 || 0.1779123 ||align=right|      4.6 : 1
 * align=right| 279,552 || 0.1632607 ||align=right|      5.1 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|  26,880 || 0.0156981 ||align=right|     62.7 : 1
 * align=right|  53,760 || 0.0313963 ||align=right|     30.9 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 3}{4 \choose 1}^3{49-4r \choose 1} \end{matrix}$$
 * align=right| 114,240 || 0.0667171 ||align=right|     14.0 : 1
 * align=right| 139,776 || 0.0816304 ||align=right|     11.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  30,240 || 0.0176604 ||align=right|     55.6 : 1
 * align=right|  48,384 || 0.0282567 ||align=right|     34.4 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1} \end{matrix}$$
 * align=right|  51,408 || 0.0300227 ||align=right|     32.3 : 1
 * align=right|  52,416 || 0.0306114 ||align=right|     31.7 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,016 || 0.0011774 ||align=right|    848.4 : 1
 * align=right|   2,688 || 0.0015698 ||align=right|    636.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|   2,268 || 0.0013245 ||align=right|    754.0 : 1
 * align=right|   3,024 || 0.0017660 ||align=right|    565.2 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{49-4r \choose 2} \end{matrix}$$
 * align=right| 137,088 || 0.0800605 ||align=right|     11.5 : 1
 * align=right| 104,832 || 0.0612228 ||align=right|     15.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   6,720 || 0.0039245 ||align=right|    253.8 : 1
 * align=right|  10,752 || 0.0062793 ||align=right|    158.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 2}{4 \choose 1}^2{49-4r \choose 1} \end{matrix}$$
 * align=right|  17,136 || 0.0100076 ||align=right|     98.9 : 1
 * align=right|  17,472 || 0.0102038 ||align=right|     97.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   3,024 || 0.0017660 ||align=right|    565.2 : 1
 * align=right|   4,032 || 0.0023547 ||align=right|    423.7 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     336 || 0.0001962 ||align=right|  5,095.1 : 1
 * align=right|     448 || 0.0002616 ||align=right|  3,821.1 : 1
 * colspan=2| Total
 * align=right|1,116,780 || 0.6522090 ||align=right|     0.5 : 1
 * align=right|1,359,568 || 0.7939992 ||align=right|     0.3 : 1
 * align=right|1,359,568 || 0.7939992 ||align=right|     0.3 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     168 || 0.0000981 ||align=right| 10,191.3 : 1
 * align=right|     224 || 0.0001308 ||align=right|  7,643.2 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 4}{49-4r \choose 1} \end{matrix}$$
 * align=right|     119 || 0.0000695 ||align=right| 14,388.1 : 1
 * align=right|     104 || 0.0000607 ||align=right| 16,463.5 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * align=right|   1,344 || 0.0007849 ||align=right|  1,273.0 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 1} \end{matrix}$$
 * align=right|  11,424 || 0.0066717 ||align=right|    148.9 : 1
 * align=right|  11,648 || 0.0068025 ||align=right|    146.0 : 1
 * $$\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{49-4r \choose 1} \end{matrix}$$
 * align=right|  12,852 || 0.0075057 ||align=right|    132.2 : 1
 * align=right|  13,104 || 0.0076528 ||align=right|    129.7 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{49-4r \choose 2} \end{matrix}$$
 * align=right|   3,808 || 0.0022239 ||align=right|    448.7 : 1
 * align=right|   2,496 || 0.0014577 ||align=right|    685.0 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{49-4r \choose 2} \end{matrix}$$
 * align=right| 137,088 || 0.0800605 ||align=right|     11.5 : 1
 * align=right| 104,832 || 0.0612228 ||align=right|     15.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      42 || 0.0000245 ||align=right| 40,768.1 : 1
 * align=right|      48 || 0.0000280 ||align=right| 35,672.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 1} \end{matrix}$$
 * align=right|     476 || 0.0002780 ||align=right|  3,596.3 : 1
 * align=right|     416 || 0.0002429 ||align=right|  4,115.1 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{49-4r \choose 2} \end{matrix}$$
 * align=right|     136 || 0.0000794 ||align=right| 12,589.5 : 1
 * align=right|      78 || 0.0000456 ||align=right| 21,951.6 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      84 || 0.0000491 ||align=right| 20,383.6 : 1
 * align=right|      96 || 0.0000561 ||align=right| 17,835.5 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 1} \end{matrix}$$
 * align=right|   2,142 || 0.0012509 ||align=right|    798.4 : 1
 * align=right|   1,872 || 0.0010933 ||align=right|    913.7 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-1 \choose 1}{4 \choose 1}{49-4r \choose 2} \end{matrix}$$
 * align=right|  11,424 || 0.0066717 ||align=right|    148.9 : 1
 * align=right|   7,488 || 0.0043731 ||align=right|    227.7 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|      21 || 0.0000123 ||align=right| 81,537.3 : 1
 * align=right|      24 || 0.0000140 ||align=right| 71,345.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{49-4r \choose 1} \end{matrix}$$
 * align=right|   1,428 || 0.0008340 ||align=right|  1,198.1 : 1
 * align=right|   1,248 || 0.0007288 ||align=right|  1,371.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{49-4r \choose 2} \end{matrix}$$
 * align=right|  17,136 || 0.0100076 ||align=right|     98.9 : 1
 * align=right|  11,232 || 0.0065596 ||align=right|    151.4 : 1
 * $$\begin{matrix} {4r-1 \choose 2}{49-4r \choose 3} \end{matrix}$$
 * align=right| 316,200 || 0.1846635 ||align=right|      4.4 : 1
 * align=right| 170,170 || 0.0993807 ||align=right|      9.1 : 1
 * $$\begin{matrix} {4r-1 \choose 1}{49-4r \choose 4} \end{matrix}$$
 * align=right|  73,780 || 0.0430881 ||align=right|     22.2 : 1
 * align=right|  25,025 || 0.0146148 ||align=right|     67.4 : 1
 * $$\begin{matrix} {49-4r \choose 5} \end{matrix}$$
 * align=right|   6,188 || 0.0036138 ||align=right|    275.7 : 1
 * align=right|   1,287 || 0.0007516 ||align=right|  1,329.5 : 1
 * colspan=2| Total
 * align=right| 595,524 || 0.3477910 ||align=right|      1.9 : 1
 * align=right| 352,736 || 0.2060008 ||align=right|      3.9 : 1
 * }
 * }

Derivations for making low hands with a low pair and two high cards
Starting hands with a low pair and two high cards (xx**) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With two high cards in the hand, there are $$52 - 4r - 2 = 50 - 4r$$ high cards and $$4r - 4 + 2 = 4r - 2$$ low cards left in the deck. The following tables show the derivations when holding a low pair and two high cards.


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=3| Miss low ! ABC ! xAB ! rowspan=8| No low ! AAA ! AAB ! xxA ! xAA ! ??* ! ?** ! ***
 * +Derivations for low hand shape xx** (a low pair and two high cards) on the flop
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   2,240 || 0.1295097 ||align=right|      6.7 : 1
 * align=right|   3,584 || 0.2072155 ||align=right|      3.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     672 || 0.0388529 ||align=right|     24.7 : 1
 * align=right|     896 || 0.0518039 ||align=right|     18.3 : 1
 * colspan=2| Total
 * align=right|   2,912 || 0.1683626 ||align=right|      4.9 : 1
 * align=right|   4,480 || 0.2590194 ||align=right|      2.9 : 1
 * align=right|   4,480 || 0.2590194 ||align=right|      2.9 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      28 || 0.0016189 ||align=right|    616.7 : 1
 * align=right|      32 || 0.0018501 ||align=right|    539.5 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,008 || 0.0582794 ||align=right|     16.2 : 1
 * align=right|   1,344 || 0.0777058 ||align=right|     11.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      28 || 0.0016189 ||align=right|    616.7 : 1
 * align=right|      32 || 0.0018501 ||align=right|    539.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      84 || 0.0048566 ||align=right|    204.9 : 1
 * align=right|      96 || 0.0055504 ||align=right|    179.2 : 1
 * $$\begin{matrix} {4r-2 \choose 2}{50-4r \choose 1} \end{matrix}$$
 * align=right|   7,830 || 0.4527058 ||align=right|      1.2 : 1
 * align=right|   7,854 || 0.4540934 ||align=right|      1.2 : 1
 * $$\begin{matrix} {4r-2 \choose 1}{50-4r \choose 2} \end{matrix}$$
 * align=right|   4,590 || 0.2653793 ||align=right|      2.8 : 1
 * align=right|   3,094 || 0.1788853 ||align=right|      4.6 : 1
 * $$\begin{matrix} {50-4r \choose 3} \end{matrix}$$
 * align=right|     816 || 0.0471785 ||align=right|     20.2 : 1
 * align=right|     364 || 0.0210453 ||align=right|     46.5 : 1
 * colspan=2| Total
 * align=right|  14,384 || 0.8316374 ||align=right|      0.2 : 1
 * align=right|  12,816 || 0.7409806 ||align=right|      0.3 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=8| Miss low ! ABCD ! AABC ! ABC* ! xxAB ! xABC ! xAAB ! xAB* ! rowspan=13| No low ! AAAA ! AAAB ! AABB ! AAA* ! AAB* ! xxAA ! xxA* ! xAAA ! xAA* ! ??** ! ?*** ! ****
 * +Derivations for low hand shape xx** (a low pair and two high cards) on the turn
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   8,960 || 0.0460479 ||align=right|     20.7 : 1
 * align=right|  17,920 || 0.0920958 ||align=right|      9.9 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  10,080 || 0.0518039 ||align=right|     18.3 : 1
 * align=right|  16,128 || 0.0828862 ||align=right|     11.1 : 1
 * $$\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{50-4r \choose 1} \end{matrix}$$
 * align=right|  40,320 || 0.2072155 ||align=right|      3.8 : 1
 * align=right|  50,176 || 0.2578682 ||align=right|      2.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     336 || 0.0017268 ||align=right|    578.1 : 1
 * align=right|     448 || 0.0023024 ||align=right|    433.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   4,480 || 0.0230239 ||align=right|     42.4 : 1
 * align=right|   7,168 || 0.0368383 ||align=right|     26.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,016 || 0.0103608 ||align=right|     95.5 : 1
 * align=right|   2,688 || 0.0138144 ||align=right|     71.4 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$$
 * align=right|  12,096 || 0.0621647 ||align=right|     15.1 : 1
 * align=right|  12,544 || 0.0644671 ||align=right|     14.5 : 1
 * colspan=2| Total
 * align=right|  78,288 || 0.4023435 ||align=right|      1.5 : 1
 * align=right| 107,072 || 0.5502724 ||align=right|      0.8 : 1
 * align=right| 107,072 || 0.5502724 ||align=right|      0.8 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       7 || 0.0000360 ||align=right| 27,796.1 : 1
 * align=right|       8 || 0.0000411 ||align=right| 24,321.5 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     672 || 0.0034536 ||align=right|    288.6 : 1
 * align=right|     896 || 0.0046048 ||align=right|    216.2 : 1
 * $$\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     756 || 0.0038853 ||align=right|    256.4 : 1
 * align=right|   1,008 || 0.0051804 ||align=right|    192.0 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{50-4r \choose 1} \end{matrix}$$
 * align=right|     504 || 0.0025902 ||align=right|    385.1 : 1
 * align=right|     448 || 0.0023024 ||align=right|    433.3 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|  18,144 || 0.0932470 ||align=right|      9.7 : 1
 * align=right|  18,816 || 0.0967006 ||align=right|      9.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      42 || 0.0002158 ||align=right|  4,631.9 : 1
 * align=right|      48 || 0.0002467 ||align=right|  4,052.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|     504 || 0.0025902 ||align=right|    385.1 : 1
 * align=right|     448 || 0.0023024 ||align=right|    433.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      56 || 0.0002878 ||align=right|  3,473.6 : 1
 * align=right|      64 || 0.0003289 ||align=right|  3,039.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 1} \end{matrix}$$
 * align=right|   1,512 || 0.0077706 ||align=right|    127.7 : 1
 * align=right|   1,344 || 0.0069072 ||align=right|    143.8 : 1
 * $$\begin{matrix} {4r-2 \choose 2}{50-4r \choose 2} \end{matrix}$$
 * align=right|  66,555 || 0.3420444 ||align=right|      1.9 : 1
 * align=right|  51,051 || 0.2623651 ||align=right|      2.8 : 1
 * $$\begin{matrix} {4r-2 \choose 1}{50-4r \choose 3} \end{matrix}$$
 * align=right|  24,480 || 0.1258094 ||align=right|      6.9 : 1
 * align=right|  12,376 || 0.0636037 ||align=right|     14.7 : 1
 * $$\begin{matrix} {50-4r \choose 4} \end{matrix}$$
 * align=right|   3,060 || 0.0157262 ||align=right|     62.6 : 1
 * align=right|   1,001 || 0.0051444 ||align=right|    193.4 : 1
 * colspan=2| Total
 * align=right| 116,292 || 0.5976565 ||align=right|      0.7 : 1
 * align=right|  87,508 || 0.4497276 ||align=right|      1.2 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=18| Miss low ! ABCDE ! AABCD ! ABCD* ! AABBC ! AAABC ! AABC* ! ABC** ! xABCD ! xABC* ! xAABC ! xAAB* ! xAAAB ! xAABB ! xAB** ! xxABC ! xxAB* ! xxAAB ! rowspan=17| No low ! AAAAB ! AAAA* ! AAABB ! AAAB* ! AABB* ! AAA** ! AAB** ! xxAAA ! xxAA* ! xxA** ! xAAAA ! xAAA* ! xAA** ! ??*** ! ?**** ! *****
 * +Derivations for low hand shape xx** (a low pair and two high cards) on the river
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-1 \choose 5}{4 \choose 1}^5 \end{matrix}$$
 * align=right|  21,504 || 0.0125585 ||align=right|     78.6 : 1
 * align=right|  57,344 || 0.0334894 ||align=right|     28.9 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|  53,760 || 0.0313963 ||align=right|     30.9 : 1
 * align=right| 107,520 || 0.0627926 ||align=right|     14.9 : 1
 * $$\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4{50-4r \choose 1} \end{matrix}$$
 * align=right| 161,280 || 0.0941889 ||align=right|      9.6 : 1
 * align=right| 250,880 || 0.1465160 ||align=right|      5.8 : 1
 * $$\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|  15,120 || 0.0088302 ||align=right|    112.2 : 1
 * align=right|  24,192 || 0.0141283 ||align=right|     69.8 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   6,720 || 0.0039245 ||align=right|    253.8 : 1
 * align=right|  10,752 || 0.0062793 ||align=right|    158.3 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$$
 * align=right| 181,440 || 0.1059625 ||align=right|      8.4 : 1
 * align=right| 225,792 || 0.1318644 ||align=right|      6.6 : 1
 * $$\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{50-4r \choose 2} \end{matrix}$$
 * align=right| 342,720 || 0.2001514 ||align=right|      4.0 : 1
 * align=right| 326,144 || 0.1904709 ||align=right|      4.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|  17,920 || 0.0104654 ||align=right|     94.6 : 1
 * align=right|  35,840 || 0.0209309 ||align=right|     46.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 3}{4 \choose 1}^3{50-4r \choose 1} \end{matrix}$$
 * align=right|  80,640 || 0.0470944 ||align=right|     20.2 : 1
 * align=right| 100,352 || 0.0586064 ||align=right|     16.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  20,160 || 0.0117736 ||align=right|     83.9 : 1
 * align=right|  32,256 || 0.0188378 ||align=right|     52.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|  36,288 || 0.0211925 ||align=right|     46.2 : 1
 * align=right|  37,632 || 0.0219774 ||align=right|     44.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,344 || 0.0007849 ||align=right|  1,273.0 : 1
 * align=right|   1,792 || 0.0010465 ||align=right|    954.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|   1,512 || 0.0008830 ||align=right|  1,131.5 : 1
 * align=right|   2,016 || 0.0011774 ||align=right|    848.4 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{50-4r \choose 2} \end{matrix}$$
 * align=right| 102,816 || 0.0600454 ||align=right|     15.7 : 1
 * align=right|  81,536 || 0.0476177 ||align=right|     20.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   2,240 || 0.0013082 ||align=right|    763.4 : 1
 * align=right|   3,584 || 0.0020931 ||align=right|    476.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$$
 * align=right|   6,048 || 0.0035321 ||align=right|    282.1 : 1
 * align=right|   6,272 || 0.0036629 ||align=right|    272.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * align=right|   1,344 || 0.0007849 ||align=right|  1,273.0 : 1
 * colspan=2| Total
 * align=right|1,052,520 || 0.6146806 ||align=right|     0.6 : 1
 * align=right|1,305,248 || 0.7622759 ||align=right|     0.3 : 1
 * align=right|1,305,248 || 0.7622759 ||align=right|     0.3 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     168 || 0.0000981 ||align=right| 10,191.3 : 1
 * align=right|     224 || 0.0001308 ||align=right|  7,643.2 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 4}{50-4r \choose 1} \end{matrix}$$
 * align=right|     126 || 0.0000736 ||align=right| 13,588.7 : 1
 * align=right|     112 || 0.0000654 ||align=right| 15,287.4 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * align=right|   1,344 || 0.0007849 ||align=right|  1,273.0 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|  12,096 || 0.0070642 ||align=right|    140.6 : 1
 * align=right|  12,544 || 0.0073258 ||align=right|    135.5 : 1
 * $$\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{50-4r \choose 1} \end{matrix}$$
 * align=right|  13,608 || 0.0079472 ||align=right|    124.8 : 1
 * align=right|  14,112 || 0.0082415 ||align=right|    120.3 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{50-4r \choose 2} \end{matrix}$$
 * align=right|   4,284 || 0.0025019 ||align=right|    398.7 : 1
 * align=right|   2,912 || 0.0017006 ||align=right|    587.0 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2} \end{matrix}$$
 * align=right| 154,224 || 0.0900681 ||align=right|     10.1 : 1
 * align=right| 122,304 || 0.0714266 ||align=right|     13.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      28 || 0.0000164 ||align=right| 61,152.7 : 1
 * align=right|      32 || 0.0000187 ||align=right| 53,508.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 1} \end{matrix}$$
 * align=right|     756 || 0.0004415 ||align=right|  2,264.0 : 1
 * align=right|     672 || 0.0003925 ||align=right|  2,547.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-1 \choose 1}{4 \choose 1}{50-4r \choose 2} \end{matrix}$$
 * align=right|   4,284 || 0.0025019 ||align=right|    398.7 : 1
 * align=right|   2,912 || 0.0017006 ||align=right|    587.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|      14 || 0.0000082 ||align=right|122,306.4 : 1
 * align=right|      16 || 0.0000093 ||align=right|107,018.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{50-4r \choose 1} \end{matrix}$$
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * align=right|     896 || 0.0005233 ||align=right|  1,910.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{50-4r \choose 2} \end{matrix}$$
 * align=right|  12,852 || 0.0075057 ||align=right|    132.2 : 1
 * align=right|   8,736 || 0.0051019 ||align=right|    195.0 : 1
 * $$\begin{matrix} {4r-2 \choose 2}{50-4r \choose 3} \end{matrix}$$
 * align=right| 354,960 || 0.2072996 ||align=right|      3.8 : 1
 * align=right| 204,204 || 0.1192569 ||align=right|      7.4 : 1
 * $$\begin{matrix} {4r-2 \choose 1}{50-4r \choose 4} \end{matrix}$$
 * align=right|  91,800 || 0.0536120 ||align=right|     17.7 : 1
 * align=right|  34,034 || 0.0198761 ||align=right|     49.3 : 1
 * $$\begin{matrix} {50-4r \choose 5} \end{matrix}$$
 * align=right|   8,568 || 0.0050038 ||align=right|    198.8 : 1
 * align=right|   2,002 || 0.0011692 ||align=right|    854.3 : 1
 * colspan=2| Total
 * align=right| 659,784 || 0.3853194 ||align=right|      1.6 : 1
 * align=right| 407,056 || 0.2377241 ||align=right|      3.2 : 1
 * }
 * }

Derivations for making low hands with a low three of a kind and a high card
Starting hands with a low three of a kind and a high card (xxx*) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With one high card in the hand, there are $$52 - 4r - 1 = 51 - 4r$$ high cards and $$4r - 4 + 1 = 4r - 3$$ low cards left in the deck. The following tables show the derivations when holding a low three of a kind and a high card.


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=3| Miss low ! ABC ! xAB ! rowspan=7| No low ! AAA ! AAB ! xAA ! ??* ! ?** ! ***
 * +Derivations for low hand shape xxx* (a low three of a kind and a high card) on the flop
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   2,240 || 0.1295097 ||align=right|      6.7 : 1
 * align=right|   3,584 || 0.2072155 ||align=right|      3.8 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     336 || 0.0194265 ||align=right|     50.5 : 1
 * align=right|     448 || 0.0259019 ||align=right|     37.6 : 1
 * colspan=2| Total
 * align=right|   2,576 || 0.1489362 ||align=right|      5.7 : 1
 * align=right|   4,032 || 0.2331175 ||align=right|      3.3 : 1
 * align=right|   4,032 || 0.2331175 ||align=right|      3.3 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      28 || 0.0016189 ||align=right|    616.7 : 1
 * align=right|      32 || 0.0018501 ||align=right|    539.5 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,008 || 0.0582794 ||align=right|     16.2 : 1
 * align=right|   1,344 || 0.0777058 ||align=right|     11.9 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      42 || 0.0024283 ||align=right|    410.8 : 1
 * align=right|      48 || 0.0027752 ||align=right|    359.3 : 1
 * $$\begin{matrix} {4r-3 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|   7,714 || 0.4459991 ||align=right|      1.2 : 1
 * align=right|   7,920 || 0.4579093 ||align=right|      1.2 : 1
 * $$\begin{matrix} {4r-3 \choose 1}{51-4r \choose 2} \end{matrix}$$
 * align=right|   4,959 || 0.2867137 ||align=right|      2.5 : 1
 * align=right|   3,465 || 0.2003353 ||align=right|      4.0 : 1
 * $$\begin{matrix} {51-4r \choose 3} \end{matrix}$$
 * align=right|     969 || 0.0560245 ||align=right|     16.8 : 1
 * align=right|     455 || 0.0263067 ||align=right|     37.0 : 1
 * colspan=2| Total
 * align=right|  14,720 || 0.8510638 ||align=right|      0.2 : 1
 * align=right|  13,264 || 0.7668825 ||align=right|      0.3 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=7| Miss low ! ABCD ! AABC ! ABC* ! xABC ! xAAB ! xAB* ! rowspan=11| No low ! AAAA ! AAAB ! AABB ! AAA* ! AAB* ! xAAA ! xAA* ! ??** ! ?*** ! ****
 * +Derivations for low hand shape xxx* (a low three of a kind and a high card) on the turn
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   8,960 || 0.0460479 ||align=right|     20.7 : 1
 * align=right|  17,920 || 0.0920958 ||align=right|      9.9 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  10,080 || 0.0518039 ||align=right|     18.3 : 1
 * align=right|  16,128 || 0.0828862 ||align=right|     11.1 : 1
 * $$\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$$
 * align=right|  42,560 || 0.2187275 ||align=right|      3.6 : 1
 * align=right|  53,760 || 0.2762874 ||align=right|      2.6 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   2,240 || 0.0115120 ||align=right|     85.9 : 1
 * align=right|   3,584 || 0.0184192 ||align=right|     53.3 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,008 || 0.0051804 ||align=right|    192.0 : 1
 * align=right|   1,344 || 0.0069072 ||align=right|    143.8 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|   6,384 || 0.0328091 ||align=right|     29.5 : 1
 * align=right|   6,720 || 0.0345359 ||align=right|     28.0 : 1
 * colspan=2| Total
 * align=right|  71,232 || 0.3660808 ||align=right|      1.7 : 1
 * align=right|  99,456 || 0.5111317 ||align=right|      1.0 : 1
 * align=right|  99,456 || 0.5111317 ||align=right|      1.0 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       7 || 0.0000360 ||align=right| 27,796.1 : 1
 * align=right|       8 || 0.0000411 ||align=right| 24,321.5 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     672 || 0.0034536 ||align=right|    288.6 : 1
 * align=right|     896 || 0.0046048 ||align=right|    216.2 : 1
 * $$\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     756 || 0.0038853 ||align=right|    256.4 : 1
 * align=right|   1,008 || 0.0051804 ||align=right|    192.0 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$$
 * align=right|     532 || 0.0027341 ||align=right|    364.8 : 1
 * align=right|     480 || 0.0024669 ||align=right|    404.4 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|  19,152 || 0.0984274 ||align=right|      9.2 : 1
 * align=right|  20,160 || 0.1036078 ||align=right|      8.7 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      28 || 0.0001439 ||align=right|  6,948.3 : 1
 * align=right|      32 || 0.0001645 ||align=right|  6,079.6 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|     798 || 0.0041011 ||align=right|    242.8 : 1
 * align=right|     720 || 0.0037003 ||align=right|    269.3 : 1
 * $$\begin{matrix} {4r-3 \choose 2}{51-4r \choose 2} \end{matrix}$$
 * align=right|  69,426 || 0.3567993 ||align=right|      1.8 : 1
 * align=right|  55,440 || 0.2849214 ||align=right|      2.5 : 1
 * $$\begin{matrix} {4r-3 \choose 1}{51-4r \choose 3} \end{matrix}$$
 * align=right|  28,101 || 0.1444187 ||align=right|      5.9 : 1
 * align=right|  15,015 || 0.0771662 ||align=right|     12.0 : 1
 * $$\begin{matrix} {51-4r \choose 4} \end{matrix}$$
 * align=right|   3,876 || 0.0199198 ||align=right|     49.2 : 1
 * align=right|   1,365 || 0.0070151 ||align=right|    141.5 : 1
 * colspan=2| Total
 * align=right| 123,348 || 0.6339192 ||align=right|      0.6 : 1
 * align=right|  95,124 || 0.4888683 ||align=right|      1.0 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=15| Miss low ! ABCDE ! AABCD ! ABCD* ! AABBC ! AAABC ! AABC* ! ABC** ! xABCD ! xABC* ! xAABC ! xAAB* ! xAAAB ! xAABB ! xAB** ! rowspan=14| No low ! AAAAB ! AAAA* ! AAABB ! AAAB* ! AABB* ! AAA** ! AAB** ! xAAAA ! xAAA* ! xAA** ! ??*** ! ?**** ! *****
 * +Derivations for low hand shape xxx* (a low three of a kind and a high card) on the river
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-1 \choose 5}{4 \choose 1}^5 \end{matrix}$$
 * align=right|  21,504 || 0.0125585 ||align=right|     78.6 : 1
 * align=right|  57,344 || 0.0334894 ||align=right|     28.9 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|  53,760 || 0.0313963 ||align=right|     30.9 : 1
 * align=right| 107,520 || 0.0627926 ||align=right|     14.9 : 1
 * $$\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4{51-4r \choose 1} \end{matrix}$$
 * align=right| 170,240 || 0.0994216 ||align=right|      9.1 : 1
 * align=right| 268,800 || 0.1569815 ||align=right|      5.4 : 1
 * $$\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|  15,120 || 0.0088302 ||align=right|    112.2 : 1
 * align=right|  24,192 || 0.0141283 ||align=right|     69.8 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   6,720 || 0.0039245 ||align=right|    253.8 : 1
 * align=right|  10,752 || 0.0062793 ||align=right|    158.3 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right| 191,520 || 0.1118493 ||align=right|      7.9 : 1
 * align=right| 241,920 || 0.1412833 ||align=right|      6.1 : 1
 * $$\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{51-4r \choose 2} \end{matrix}$$
 * align=right| 383,040 || 0.2236986 ||align=right|      3.5 : 1
 * align=right| 376,320 || 0.2197741 ||align=right|      3.6 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   8,960 || 0.0052327 ||align=right|    190.1 : 1
 * align=right|  17,920 || 0.0104654 ||align=right|     94.6 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$$
 * align=right|  42,560 || 0.0248554 ||align=right|     39.2 : 1
 * align=right|  53,760 || 0.0313963 ||align=right|     30.9 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  10,080 || 0.0058868 ||align=right|    168.9 : 1
 * align=right|  16,128 || 0.0094189 ||align=right|    105.2 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|  19,152 || 0.0111849 ||align=right|     88.4 : 1
 * align=right|  20,160 || 0.0117736 ||align=right|     83.9 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     672 || 0.0003925 ||align=right|  2,547.1 : 1
 * align=right|     896 || 0.0005233 ||align=right|  1,910.1 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     756 || 0.0004415 ||align=right|  2,264.0 : 1
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{51-4r \choose 2} \end{matrix}$$
 * align=right|  57,456 || 0.0335548 ||align=right|     28.8 : 1
 * align=right|  47,040 || 0.0274718 ||align=right|     35.4 : 1
 * colspan=2| Total
 * align=right| 981,540 || 0.5732277 ||align=right|      0.7 : 1
 * align=right|1,243,760 || 0.7263663 ||align=right|     0.4 : 1
 * align=right|1,243,760 || 0.7263663 ||align=right|     0.4 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     168 || 0.0000981 ||align=right| 10,191.3 : 1
 * align=right|     224 || 0.0001308 ||align=right|  7,643.2 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 4}{51-4r \choose 1} \end{matrix}$$
 * align=right|     133 || 0.0000777 ||align=right| 12,873.5 : 1
 * align=right|     120 || 0.0000701 ||align=right| 14,268.2 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * align=right|   1,344 || 0.0007849 ||align=right|  1,273.0 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|  12,768 || 0.0074566 ||align=right|    133.1 : 1
 * align=right|  13,440 || 0.0078491 ||align=right|    126.4 : 1
 * $$\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|  14,364 || 0.0083887 ||align=right|    118.2 : 1
 * align=right|  15,120 || 0.0088302 ||align=right|    112.2 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{51-4r \choose 2} \end{matrix}$$
 * align=right|   4,788 || 0.0027962 ||align=right|    356.6 : 1
 * align=right|   3,360 || 0.0019623 ||align=right|    508.6 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$$
 * align=right| 172,368 || 0.1006644 ||align=right|      8.9 : 1
 * align=right| 141,120 || 0.0824153 ||align=right|     11.1 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       7 || 0.0000041 ||align=right|244,613.9 : 1
 * align=right|       8 || 0.0000047 ||align=right|214,037.0 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$$
 * align=right|     532 || 0.0003107 ||align=right|  3,217.6 : 1
 * align=right|     480 || 0.0002803 ||align=right|  3,566.3 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{51-4r \choose 2} \end{matrix}$$
 * align=right|   7,182 || 0.0041943 ||align=right|    237.4 : 1
 * align=right|   5,040 || 0.0029434 ||align=right|    338.7 : 1
 * $$\begin{matrix} {4r-3 \choose 2}{51-4r \choose 3} \end{matrix}$$
 * align=right| 393,414 || 0.2297571 ||align=right|      3.4 : 1
 * align=right| 240,240 || 0.1403022 ||align=right|      6.1 : 1
 * $$\begin{matrix} {4r-3 \choose 1}{51-4r \choose 4} \end{matrix}$$
 * align=right| 112,404 || 0.0656449 ||align=right|     14.2 : 1
 * align=right|  45,045 || 0.0263067 ||align=right|     37.0 : 1
 * $$\begin{matrix} {51-4r \choose 5} \end{matrix}$$
 * align=right|  11,628 || 0.0067909 ||align=right|    146.3 : 1
 * align=right|   3,003 || 0.0017538 ||align=right|    569.2 : 1
 * colspan=2| Total
 * align=right| 730,764 || 0.4267723 ||align=right|      1.3 : 1
 * align=right| 468,544 || 0.2736337 ||align=right|      2.7 : 1
 * }
 * }

Derivations for making low hands with a low four of a kind
Starting hands with a low four of a kind (xxxx) cannot make a low hand, so the only possibilities are having a low hand possible on the board or not. With no high cards in the hand, there are $$52 - 4r - 0 = 52 - 4r$$ high cards and $$4r - 4 + 0 = 4r - 4$$ low cards left in the deck. The following tables show the derivations when holding a low four of a kind.


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=1| Miss low ! ABC ! rowspan=6| No low ! AAA ! AAB ! ??* ! ?** ! ***
 * +Derivations for low hand shape xxxx (a low four of a kind) on the flop
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   2,240 || 0.1295097 ||align=right|      6.7 : 1
 * align=right|   3,584 || 0.2072155 ||align=right|      3.8 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      28 || 0.0016189 ||align=right|    616.7 : 1
 * align=right|      32 || 0.0018501 ||align=right|    539.5 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,008 || 0.0582794 ||align=right|     16.2 : 1
 * align=right|   1,344 || 0.0777058 ||align=right|     11.9 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   7,560 || 0.4370953 ||align=right|      1.3 : 1
 * align=right|   7,936 || 0.4588344 ||align=right|      1.2 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|   5,320 || 0.3075856 ||align=right|      2.3 : 1
 * align=right|   3,840 || 0.2220167 ||align=right|      3.5 : 1
 * $$\begin{matrix} {52-4r \choose 3} \end{matrix}$$
 * align=right|   1,140 || 0.0659112 ||align=right|     14.2 : 1
 * align=right|     560 || 0.0323774 ||align=right|     29.9 : 1
 * colspan=2| Total
 * align=right|  15,056 || 0.8704903 ||align=right|      0.1 : 1
 * align=right|  13,712 || 0.7927845 ||align=right|      0.3 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=4| Miss low ! ABCD ! AABC ! ABC* ! rowspan=9| No low ! AAAA ! AAAB ! AABB ! AAA* ! AAB* ! ??** ! ?*** ! ****
 * +Derivations for low hand shape xxxx (a low four of a kind) on the turn
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   8,960 || 0.0460479 ||align=right|     20.7 : 1
 * align=right|  17,920 || 0.0920958 ||align=right|      9.9 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  10,080 || 0.0518039 ||align=right|     18.3 : 1
 * align=right|  16,128 || 0.0828862 ||align=right|     11.1 : 1
 * $$\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$$
 * align=right|  44,800 || 0.2302395 ||align=right|      3.3 : 1
 * align=right|  57,344 || 0.2947065 ||align=right|      2.4 : 1
 * colspan=2| Total
 * align=right|  63,840 || 0.3280913 ||align=right|      2.0 : 1
 * align=right|  91,392 || 0.4696886 ||align=right|      1.1 : 1
 * align=right|  91,392 || 0.4696886 ||align=right|      1.1 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       7 || 0.0000360 ||align=right| 27,796.1 : 1
 * align=right|       8 || 0.0000411 ||align=right| 24,321.5 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     672 || 0.0034536 ||align=right|    288.6 : 1
 * align=right|     896 || 0.0046048 ||align=right|    216.2 : 1
 * $$\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     756 || 0.0038853 ||align=right|    256.4 : 1
 * align=right|   1,008 || 0.0051804 ||align=right|    192.0 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|     560 || 0.0028780 ||align=right|    346.5 : 1
 * align=right|     512 || 0.0026313 ||align=right|    379.0 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|  20,160 || 0.1036078 ||align=right|      8.7 : 1
 * align=right|  21,504 || 0.1105150 ||align=right|      8.0 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 2} \end{matrix}$$
 * align=right|  71,820 || 0.3691027 ||align=right|      1.7 : 1
 * align=right|  59,520 || 0.3058896 ||align=right|      2.3 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 3} \end{matrix}$$
 * align=right|  31,920 || 0.1640456 ||align=right|      5.1 : 1
 * align=right|  17,920 || 0.0920958 ||align=right|      9.9 : 1
 * $$\begin{matrix} {52-4r \choose 4} \end{matrix}$$
 * align=right|   4,845 || 0.0248998 ||align=right|     39.2 : 1
 * align=right|   1,820 || 0.0093535 ||align=right|    105.9 : 1
 * colspan=2| Total
 * align=right| 130,740 || 0.6719087 ||align=right|      0.5 : 1
 * align=right| 103,188 || 0.5303114 ||align=right|      0.9 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=8| Miss low ! ABCDE ! AABCD ! ABCD* ! AABBC ! AAABC ! AABC* ! ABC** ! rowspan=11| No low ! AAAAB ! AAAA* ! AAABB ! AAAB* ! AABB* ! AAA** ! AAB** ! ??*** ! ?**** ! *****
 * +Derivations for low hand shape xxxx (a low four of a kind) on the river
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-1 \choose 5}{4 \choose 1}^5 \end{matrix}$$
 * align=right|  21,504 || 0.0125585 ||align=right|     78.6 : 1
 * align=right|  57,344 || 0.0334894 ||align=right|     28.9 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|  53,760 || 0.0313963 ||align=right|     30.9 : 1
 * align=right| 107,520 || 0.0627926 ||align=right|     14.9 : 1
 * $$\begin{matrix} {r-1 \choose 4}{4 \choose 1}^4{52-4r \choose 1} \end{matrix}$$
 * align=right| 179,200 || 0.1046543 ||align=right|      8.6 : 1
 * align=right| 286,720 || 0.1674469 ||align=right|      5.0 : 1
 * $$\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|  15,120 || 0.0088302 ||align=right|    112.2 : 1
 * align=right|  24,192 || 0.0141283 ||align=right|     69.8 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   6,720 || 0.0039245 ||align=right|    253.8 : 1
 * align=right|  10,752 || 0.0062793 ||align=right|    158.3 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right| 201,600 || 0.1177361 ||align=right|      7.5 : 1
 * align=right| 258,048 || 0.1507022 ||align=right|      5.6 : 1
 * $$\begin{matrix} {r-1 \choose 3}{4 \choose 1}^3{52-4r \choose 2} \end{matrix}$$
 * align=right| 425,600 || 0.2485540 ||align=right|      3.0 : 1
 * align=right| 430,080 || 0.2511704 ||align=right|      3.0 : 1
 * colspan=2| Total
 * align=right| 903,504 || 0.5276540 ||align=right|      0.9 : 1
 * align=right|1,174,656 || 0.6860090 ||align=right|     0.5 : 1
 * align=right|1,174,656 || 0.6860090 ||align=right|     0.5 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 4}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     168 || 0.0000981 ||align=right| 10,191.3 : 1
 * align=right|     224 || 0.0001308 ||align=right|  7,643.2 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 4}{52-4r \choose 1} \end{matrix}$$
 * align=right|     140 || 0.0000818 ||align=right| 12,229.7 : 1
 * align=right|     128 || 0.0000748 ||align=right| 13,376.4 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * align=right|   1,344 || 0.0007849 ||align=right|  1,273.0 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|  13,440 || 0.0078491 ||align=right|    126.4 : 1
 * align=right|  14,336 || 0.0083723 ||align=right|    118.4 : 1
 * $$\begin{matrix} {r-1 \choose 2}{4 \choose 2}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  15,120 || 0.0088302 ||align=right|    112.2 : 1
 * align=right|  16,128 || 0.0094189 ||align=right|    105.2 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 3}{52-4r \choose 2} \end{matrix}$$
 * align=right|   5,320 || 0.0031069 ||align=right|    320.9 : 1
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * $$\begin{matrix} {r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right| 191,520 || 0.1118493 ||align=right|      7.9 : 1
 * align=right| 161,280 || 0.0941889 ||align=right|      9.6 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 3} \end{matrix}$$
 * align=right| 430,920 || 0.2516609 ||align=right|      3.0 : 1
 * align=right| 277,760 || 0.1622142 ||align=right|      5.2 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 4} \end{matrix}$$
 * align=right| 135,660 || 0.0792266 ||align=right|     11.6 : 1
 * align=right|  58,240 || 0.0340127 ||align=right|     28.4 : 1
 * $$\begin{matrix} {52-4r \choose 5} \end{matrix}$$
 * align=right|  15,504 || 0.0090545 ||align=right|    109.4 : 1
 * align=right|   4,368 || 0.0025509 ||align=right|    391.0 : 1
 * colspan=2| Total
 * align=right| 808,800 || 0.4723460 ||align=right|      1.1 : 1
 * align=right| 537,648 || 0.3139910 ||align=right|      2.2 : 1
 * }
 * }

Derivations for making low hands with two low ranks and two high cards
Starting hands with two low ranks and two high cards (xy**) can make a low hand when at least three other low ranks beside x and y appear on the board. With two high cards in the hand, there are $$52 - 4r - 2 = 50 - 4r$$ high cards and $$4r - 4 + 2 = 4r - 2$$ low cards left in the deck. The following tables show the derivations when holding two low ranks and two high cards.


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=1| Make low ! ABC ! rowspan=3| Miss low ! xAB ! xyA ! rowspan=10| No low ! AAA ! AAB ! xxx ! xxy ! xxA ! xAA ! ??* ! ?** ! ***
 * +Derivations for low hand shape xy** (two low ranks and two high cards) on the flop
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   1,280 || 0.0740056 ||align=right|     12.5 : 1
 * align=right|   2,240 || 0.1295097 ||align=right|      6.7 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,440 || 0.0832562 ||align=right|     11.0 : 1
 * align=right|   2,016 || 0.1165587 ||align=right|      7.6 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     216 || 0.0124884 ||align=right|     79.1 : 1
 * align=right|     252 || 0.0145698 ||align=right|     67.6 : 1
 * colspan=2| Total
 * align=right|   1,656 || 0.0957447 ||align=right|      9.4 : 1
 * align=right|   2,268 || 0.1311286 ||align=right|      6.6 : 1
 * align=right|   2,268 || 0.1311286 ||align=right|      6.6 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      24 || 0.0013876 ||align=right|    719.7 : 1
 * align=right|      28 || 0.0016189 ||align=right|    616.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     720 || 0.0416281 ||align=right|     23.0 : 1
 * align=right|   1,008 || 0.0582794 ||align=right|     16.2 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3} \end{matrix}$$
 * align=right|       2 || 0.0001156 ||align=right|  8,647.0 : 1
 * align=right|       2 || 0.0001156 ||align=right|  8,647.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|      18 || 0.0010407 ||align=right|    959.9 : 1
 * align=right|      18 || 0.0010407 ||align=right|    959.9 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     144 || 0.0083256 ||align=right|    119.1 : 1
 * align=right|     168 || 0.0097132 ||align=right|    102.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     216 || 0.0124884 ||align=right|     79.1 : 1
 * align=right|     252 || 0.0145698 ||align=right|     67.6 : 1
 * $$\begin{matrix} {4r-2 \choose 2}{50-4r \choose 1} \end{matrix}$$
 * align=right|   7,830 || 0.4527058 ||align=right|      1.2 : 1
 * align=right|   7,854 || 0.4540934 ||align=right|      1.2 : 1
 * $$\begin{matrix} {4r-2 \choose 1}{50-4r \choose 2} \end{matrix}$$
 * align=right|   4,590 || 0.2653793 ||align=right|      2.8 : 1
 * align=right|   3,094 || 0.1788853 ||align=right|      4.6 : 1
 * $$\begin{matrix} {50-4r \choose 3} \end{matrix}$$
 * align=right|     816 || 0.0471785 ||align=right|     20.2 : 1
 * align=right|     364 || 0.0210453 ||align=right|     46.5 : 1
 * colspan=2| Total
 * align=right|  14,360 || 0.8302498 ||align=right|      0.2 : 1
 * align=right|  12,788 || 0.7393617 ||align=right|      0.4 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=5| Make low ! ABCD ! AABC ! ABC* ! xABC ! rowspan=8| Miss low ! xxyA ! xxAB ! xyAA ! xAAB ! xyAB ! xyA* ! xAB* ! rowspan=18| No low ! AAAA ! AAAB ! AABB ! AAA* ! AAB* ! xxxy ! xxxA ! xxx* ! xxyy ! xxy* ! xxAA ! xxA* ! xAAA ! xAA* ! ??** ! ?*** ! ****
 * +Derivations for low hand shape xy** (two low ranks and two high cards) on the turn
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   3,840 || 0.0197348 ||align=right|     49.7 : 1
 * align=right|   8,960 || 0.0460479 ||align=right|     20.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   5,760 || 0.0296022 ||align=right|     32.8 : 1
 * align=right|  10,080 || 0.0518039 ||align=right|     18.3 : 1
 * $$\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{50-4r \choose 1} \end{matrix}$$
 * align=right|  23,040 || 0.1184089 ||align=right|      7.4 : 1
 * align=right|  31,360 || 0.1611676 ||align=right|      5.2 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   7,680 || 0.0394696 ||align=right|     24.3 : 1
 * align=right|  13,440 || 0.0690718 ||align=right|     13.5 : 1
 * colspan=2| Total
 * align=right|  40,320 || 0.2072155 ||align=right|      3.8 : 1
 * align=right|  63,840 || 0.3280913 ||align=right|      2.0 : 1
 * align=right|  63,840 || 0.3280913 ||align=right|      2.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     432 || 0.0022202 ||align=right|    449.4 : 1
 * align=right|     504 || 0.0025902 ||align=right|    385.1 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,440 || 0.0074006 ||align=right|    134.1 : 1
 * align=right|   2,016 || 0.0103608 ||align=right|     95.5 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     324 || 0.0016651 ||align=right|    599.6 : 1
 * align=right|     378 || 0.0019426 ||align=right|    513.8 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   4,320 || 0.0222017 ||align=right|     44.0 : 1
 * align=right|   6,048 || 0.0310823 ||align=right|     31.2 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   2,160 || 0.0111008 ||align=right|     89.1 : 1
 * align=right|   3,024 || 0.0155412 ||align=right|     63.3 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|   3,888 || 0.0199815 ||align=right|     49.0 : 1
 * align=right|   3,528 || 0.0181314 ||align=right|     54.2 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$$
 * align=right|  25,920 || 0.1332100 ||align=right|      6.5 : 1
 * align=right|  28,224 || 0.1450509 ||align=right|      5.9 : 1
 * colspan=2| Total
 * align=right|  38,484 || 0.1977798 ||align=right|      4.1 : 1
 * align=right|  43,722 || 0.2246994 ||align=right|      3.5 : 1
 * align=right|  43,722 || 0.2246994 ||align=right|      3.5 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       6 || 0.0000308 ||align=right| 32,429.0 : 1
 * align=right|       7 || 0.0000360 ||align=right| 27,796.1 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     480 || 0.0024669 ||align=right|    404.4 : 1
 * align=right|     672 || 0.0034536 ||align=right|    288.6 : 1
 * $$\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     540 || 0.0027752 ||align=right|    359.3 : 1
 * align=right|     756 || 0.0038853 ||align=right|    256.4 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{50-4r \choose 1} \end{matrix}$$
 * align=right|     432 || 0.0022202 ||align=right|    449.4 : 1
 * align=right|     392 || 0.0020146 ||align=right|    495.4 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|  14,112 || 0.0725254 ||align=right|     12.8 : 1
 * align=right|  12,960 || 0.0666050 ||align=right|     14.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|       6 || 0.0000308 ||align=right| 32,429.0 : 1
 * align=right|       6 || 0.0000308 ||align=right| 32,429.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      56 || 0.0002878 ||align=right|  3,473.6 : 1
 * align=right|      48 || 0.0002467 ||align=right|  4,052.8 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{50-4r \choose 1} \end{matrix}$$
 * align=right|      28 || 0.0001439 ||align=right|  6,948.3 : 1
 * align=right|      36 || 0.0001850 ||align=right|  5,404.0 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 2}^2 \end{matrix}$$
 * align=right|       9 || 0.0000463 ||align=right| 21,619.0 : 1
 * align=right|       9 || 0.0000463 ||align=right| 21,619.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|     324 || 0.0016651 ||align=right|    599.6 : 1
 * align=right|     252 || 0.0012951 ||align=right|    771.1 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     216 || 0.0011101 ||align=right|    899.8 : 1
 * align=right|     252 || 0.0012951 ||align=right|    771.1 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|   2,592 || 0.0133210 ||align=right|     74.1 : 1
 * align=right|   2,352 || 0.0120876 ||align=right|     81.7 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     144 || 0.0007401 ||align=right|  1,350.3 : 1
 * align=right|     168 || 0.0008634 ||align=right|  1,157.2 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{50-4r \choose 1} \end{matrix}$$
 * align=right|   3,888 || 0.0199815 ||align=right|     49.0 : 1
 * align=right|   3,528 || 0.0181314 ||align=right|     54.2 : 1
 * $$\begin{matrix} {4r-2 \choose 2}{50-4r \choose 2} \end{matrix}$$
 * align=right|  66,555 || 0.3420444 ||align=right|      1.9 : 1
 * align=right|  51,051 || 0.2623651 ||align=right|      2.8 : 1
 * $$\begin{matrix} {4r-2 \choose 1}{50-4r \choose 3} \end{matrix}$$
 * align=right|  24,480 || 0.1258094 ||align=right|      6.9 : 1
 * align=right|  12,376 || 0.0636037 ||align=right|     14.7 : 1
 * $$\begin{matrix} {50-4r \choose 4} \end{matrix}$$
 * align=right|   3,060 || 0.0157262 ||align=right|     62.6 : 1
 * align=right|   1,001 || 0.0051444 ||align=right|    193.4 : 1
 * colspan=2| Total
 * align=right| 115,776 || 0.5950046 ||align=right|      0.7 : 1
 * align=right|  87,018 || 0.4472094 ||align=right|      1.2 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=13| Make low ! ABCDE ! AABCD ! ABCD* ! AABBC ! AABC* ! AAABC ! ABC** ! xABCD ! xABC* ! xAABC ! xxABC ! xyABC ! rowspan=18| Miss low ! xxxyA ! xxxAB ! xxyyA ! xxyAB ! xxyAA ! xxyA* ! xxAAB ! xxAB* ! xyAAA ! xyAAB ! xyAA* ! xyAB* ! xyA** ! xAAAB ! xAABB ! xAAB* ! xAB** ! rowspan=24| No low ! AAAAB ! AAAA* ! AAABB ! AAAB* ! AABB* ! AAA** ! AAB** ! xxxyy ! xxxy* ! xxxAA ! xxxA* ! xxx** ! xxyy* ! xxy** ! xxAAA ! xxAA* ! xxA** ! xAAAA ! xAAA* ! xAA** ! ??*** ! ?**** ! *****
 * +Derivations for low hand shape xy** (two low ranks and two high cards) on the river
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-2 \choose 5}{4 \choose 1}^5 \end{matrix}$$
 * align=right|   6,144 || 0.0035881 ||align=right|    277.7 : 1
 * align=right|  21,504 || 0.0125585 ||align=right|     78.6 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * align=right|  53,760 || 0.0313963 ||align=right|     30.9 : 1
 * $$\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4{50-4r \choose 1} \end{matrix}$$
 * align=right|  69,120 || 0.0403667 ||align=right|     23.8 : 1
 * align=right| 125,440 || 0.0732580 ||align=right|     12.7 : 1
 * $$\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   8,640 || 0.0050458 ||align=right|    197.2 : 1
 * align=right|  15,120 || 0.0088302 ||align=right|    112.2 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$$
 * align=right| 103,680 || 0.0605500 ||align=right|     15.5 : 1
 * align=right| 141,120 || 0.0824153 ||align=right|     11.1 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * align=right|   6,720 || 0.0039245 ||align=right|    253.8 : 1
 * $$\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{50-4r \choose 2} \end{matrix}$$
 * align=right| 195,840 || 0.1143722 ||align=right|      7.7 : 1
 * align=right| 203,840 || 0.1190443 ||align=right|      7.4 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * align=right|  53,760 || 0.0313963 ||align=right|     30.9 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3{50-4r \choose 1} \end{matrix}$$
 * align=right| 138,240 || 0.0807333 ||align=right|     11.4 : 1
 * align=right| 188,160 || 0.1098870 ||align=right|      8.1 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  34,560 || 0.0201833 ||align=right|     48.5 : 1
 * align=right|  60,480 || 0.0353208 ||align=right|     27.3 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   7,680 || 0.0044852 ||align=right|    222.0 : 1
 * align=right|  13,440 || 0.0078491 ||align=right|    126.4 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|  11,520 || 0.0067278 ||align=right|    147.6 : 1
 * align=right|  20,160 || 0.0117736 ||align=right|     83.9 : 1
 * colspan=2| Total
 * align=right| 625,344 || 0.3652062 ||align=right|      1.7 : 1
 * align=right| 903,504 || 0.5276540 ||align=right|      0.9 : 1
 * align=right| 903,504 || 0.5276540 ||align=right|      0.9 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     144 || 0.0000841 ||align=right| 11,890.0 : 1
 * align=right|     168 || 0.0000981 ||align=right| 10,191.3 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     480 || 0.0002803 ||align=right|  3,566.3 : 1
 * align=right|     672 || 0.0003925 ||align=right|  2,547.1 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 2}^2{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     216 || 0.0001261 ||align=right|  7,926.3 : 1
 * align=right|     252 || 0.0001472 ||align=right|  6,793.9 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * align=right|   6,048 || 0.0035321 ||align=right|    282.1 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     648 || 0.0003784 ||align=right|  2,641.4 : 1
 * align=right|     756 || 0.0004415 ||align=right|  2,264.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|   7,776 || 0.0045412 ||align=right|    219.2 : 1
 * align=right|   7,056 || 0.0041208 ||align=right|    241.7 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * align=right|   6,048 || 0.0035321 ||align=right|    282.1 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$$
 * align=right|  25,920 || 0.0151375 ||align=right|     65.1 : 1
 * align=right|  28,224 || 0.0164831 ||align=right|     59.7 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     216 || 0.0001261 ||align=right|  7,926.3 : 1
 * align=right|     252 || 0.0001472 ||align=right|  6,793.9 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   6,480 || 0.0037844 ||align=right|    263.2 : 1
 * align=right|   9,072 || 0.0052981 ||align=right|    187.7 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 2}{50-4r \choose 1} \end{matrix}$$
 * align=right|   5,832 || 0.0034059 ||align=right|    292.6 : 1
 * align=right|   5,292 || 0.0030906 ||align=right|    322.6 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 2}{4 \choose 1}^2{50-4r \choose 1} \end{matrix}$$
 * align=right|  38,880 || 0.0227062 ||align=right|     43.0 : 1
 * align=right|  42,336 || 0.0247246 ||align=right|     39.4 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2} \end{matrix}$$
 * align=right|  33,048 || 0.0193003 ||align=right|     50.8 : 1
 * align=right|  22,932 || 0.0133925 ||align=right|     73.7 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,880 || 0.0016819 ||align=right|    593.6 : 1
 * align=right|   4,032 || 0.0023547 ||align=right|    423.7 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|   3,240 || 0.0018922 ||align=right|    527.5 : 1
 * align=right|   4,536 || 0.0026491 ||align=right|    376.5 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|  77,760 || 0.0454125 ||align=right|     21.0 : 1
 * align=right|  84,672 || 0.0494492 ||align=right|     19.2 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{50-4r \choose 2} \end{matrix}$$
 * align=right| 220,320 || 0.1286687 ||align=right|      6.8 : 1
 * align=right| 183,456 || 0.1071399 ||align=right|      8.3 : 1
 * colspan=2| Total
 * align=right| 432,480 || 0.2525720 ||align=right|      3.0 : 1
 * align=right| 405,804 || 0.2369930 ||align=right|      3.2 : 1
 * align=right| 405,804 || 0.2369930 ||align=right|      3.2 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 4}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     120 || 0.0000701 ||align=right| 14,268.2 : 1
 * align=right|     168 || 0.0000981 ||align=right| 10,191.3 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 4}{50-4r \choose 1} \end{matrix}$$
 * align=right|     108 || 0.0000631 ||align=right| 15,853.7 : 1
 * align=right|      98 || 0.0000572 ||align=right| 17,471.5 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|   8,640 || 0.0050458 ||align=right|    197.2 : 1
 * align=right|   9,408 || 0.0054944 ||align=right|    181.0 : 1
 * $$\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{50-4r \choose 1} \end{matrix}$$
 * align=right|   9,720 || 0.0056766 ||align=right|    175.2 : 1
 * align=right|  10,584 || 0.0061811 ||align=right|    160.8 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{50-4r \choose 2} \end{matrix}$$
 * align=right|   3,672 || 0.0021445 ||align=right|    465.3 : 1
 * align=right|   2,548 || 0.0014881 ||align=right|    671.0 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2} \end{matrix}$$
 * align=right| 110,160 || 0.0643344 ||align=right|     14.5 : 1
 * align=right|  91,728 || 0.0535699 ||align=right|     17.7 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 2} \end{matrix}$$
 * align=right|       6 || 0.0000035 ||align=right|285,383.0 : 1
 * align=right|       6 || 0.0000035 ||align=right|285,383.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|     108 || 0.0000631 ||align=right| 15,853.7 : 1
 * align=right|      84 || 0.0000491 ||align=right| 20,383.6 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      72 || 0.0000420 ||align=right| 23,781.0 : 1
 * align=right|      84 || 0.0000491 ||align=right| 20,383.6 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|     864 || 0.0005046 ||align=right|  1,980.8 : 1
 * align=right|     784 || 0.0004579 ||align=right|  2,183.1 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{50-4r \choose 2} \end{matrix}$$
 * align=right|     306 || 0.0001787 ||align=right|  5,594.8 : 1
 * align=right|     182 || 0.0001063 ||align=right|  9,407.3 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 2}^2{50-4r \choose 1} \end{matrix}$$
 * align=right|     162 || 0.0000946 ||align=right| 10,568.8 : 1
 * align=right|     126 || 0.0000736 ||align=right| 13,588.7 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{50-4r \choose 2} \end{matrix}$$
 * align=right|   2,754 || 0.0016084 ||align=right|    620.8 : 1
 * align=right|   1,638 || 0.0009566 ||align=right|  1,044.4 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     144 || 0.0000841 ||align=right| 11,890.0 : 1
 * align=right|     168 || 0.0000981 ||align=right| 10,191.3 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{50-4r \choose 1} \end{matrix}$$
 * align=right|   3,888 || 0.0022706 ||align=right|    439.4 : 1
 * align=right|   3,528 || 0.0020604 ||align=right|    484.3 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 2} \end{matrix}$$
 * align=right|  22,032 || 0.0128669 ||align=right|     76.7 : 1
 * align=right|  15,288 || 0.0089283 ||align=right|    111.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|      36 || 0.0000210 ||align=right| 47,563.0 : 1
 * align=right|      42 || 0.0000245 ||align=right| 40,768.1 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}{50-4r \choose 1} \end{matrix}$$
 * align=right|   2,592 || 0.0015137 ||align=right|    659.6 : 1
 * align=right|   2,352 || 0.0013736 ||align=right|    727.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{50-4r \choose 2} \end{matrix}$$
 * align=right|  33,048 || 0.0193003 ||align=right|     50.8 : 1
 * align=right|  22,932 || 0.0133925 ||align=right|     73.7 : 1
 * $$\begin{matrix} {4r-2 \choose 2}{50-4r \choose 3} \end{matrix}$$
 * align=right| 354,960 || 0.2072996 ||align=right|      3.8 : 1
 * align=right| 204,204 || 0.1192569 ||align=right|      7.4 : 1
 * $$\begin{matrix} {4r-2 \choose 1}{50-4r \choose 4} \end{matrix}$$
 * align=right|  91,800 || 0.0536120 ||align=right|     17.7 : 1
 * align=right|  34,034 || 0.0198761 ||align=right|     49.3 : 1
 * $$\begin{matrix} {50-4r \choose 5} \end{matrix}$$
 * align=right|   8,568 || 0.0050038 ||align=right|    198.8 : 1
 * align=right|   2,002 || 0.0011692 ||align=right|    854.3 : 1
 * colspan=2| Total
 * align=right| 654,480 || 0.3822218 ||align=right|      1.6 : 1
 * align=right| 402,996 || 0.2353531 ||align=right|      3.2 : 1
 * }
 * }

Derivations for making low hands with a low pair, a low card and a high card
Starting hands with a low pair, a low card and a high card (xxy*) can make a low hand when at least three other low ranks beside x and y appear on the board. With one high card in the hand, there are $$52 - 4r - 1 = 51 - 4r$$ high cards and $$4r - 4 + 1 = 4r - 3$$ low cards left in the deck. The following tables show the derivations when holding a low pair, a low card and a high card.


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=1| Make low ! ABC ! rowspan=4| Miss low ! xAB ! yAB ! xyA ! rowspan=13| No low ! AAA ! AAB ! xxy ! xxA ! xyy ! xAA ! yyy ! yyA ! yAA ! ??* ! ?** ! ***
 * +Derivations for low hand shape xxy* (a low pair, a low card and a high card) on the flop
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   1,280 || 0.0740056 ||align=right|     12.5 : 1
 * align=right|   2,240 || 0.1295097 ||align=right|      6.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     480 || 0.0277521 ||align=right|     35.0 : 1
 * align=right|     672 || 0.0388529 ||align=right|     24.7 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     720 || 0.0416281 ||align=right|     23.0 : 1
 * align=right|   1,008 || 0.0582794 ||align=right|     16.2 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     144 || 0.0083256 ||align=right|    119.1 : 1
 * align=right|     168 || 0.0097132 ||align=right|    102.0 : 1
 * colspan=2| Total
 * align=right|   1,344 || 0.0777058 ||align=right|     11.9 : 1
 * align=right|   1,848 || 0.1068455 ||align=right|      8.4 : 1
 * align=right|   1,848 || 0.1068455 ||align=right|      8.4 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      24 || 0.0013876 ||align=right|    719.7 : 1
 * align=right|      28 || 0.0016189 ||align=right|    616.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     720 || 0.0416281 ||align=right|     23.0 : 1
 * align=right|   1,008 || 0.0582794 ||align=right|     16.2 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|       3 || 0.0001735 ||align=right|  5,764.3 : 1
 * align=right|       3 || 0.0001735 ||align=right|  5,764.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      24 || 0.0013876 ||align=right|    719.7 : 1
 * align=right|      28 || 0.0016189 ||align=right|    616.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2} \end{matrix}$$
 * align=right|       6 || 0.0003469 ||align=right|  2,881.7 : 1
 * align=right|       6 || 0.0003469 ||align=right|  2,881.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      72 || 0.0041628 ||align=right|    239.2 : 1
 * align=right|      84 || 0.0048566 ||align=right|    204.9 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3} \end{matrix}$$
 * align=right|       1 || 0.0000578 ||align=right| 17,295.0 : 1
 * align=right|       1 || 0.0000578 ||align=right| 17,295.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     108 || 0.0062442 ||align=right|    159.1 : 1
 * align=right|     126 || 0.0072849 ||align=right|    136.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     216 || 0.0124884 ||align=right|     79.1 : 1
 * align=right|     252 || 0.0145698 ||align=right|     67.6 : 1
 * $$\begin{matrix} {4r-3 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|   7,714 || 0.4459991 ||align=right|      1.2 : 1
 * align=right|   7,920 || 0.4579093 ||align=right|      1.2 : 1
 * $$\begin{matrix} {4r-3 \choose 1}{51-4r \choose 2} \end{matrix}$$
 * align=right|   4,959 || 0.2867137 ||align=right|      2.5 : 1
 * align=right|   3,465 || 0.2003353 ||align=right|      4.0 : 1
 * $$\begin{matrix} {51-4r \choose 3} \end{matrix}$$
 * align=right|     969 || 0.0560245 ||align=right|     16.8 : 1
 * align=right|     455 || 0.0263067 ||align=right|     37.0 : 1
 * colspan=2| Total
 * align=right|  14,672 || 0.8482886 ||align=right|      0.2 : 1
 * align=right|  13,208 || 0.7636448 ||align=right|      0.3 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=6| Make low ! ABCD ! AABC ! ABC* ! xABC ! yABC ! rowspan=12| Miss low ! xxyA ! xxAB ! xyyA ! xyAA ! xAAB ! xyAB ! xyA* ! xAB* ! yyAB ! yAAB ! yAB* ! rowspan=23| No low ! AAAA ! AAAB ! AABB ! AAA* ! AAB* ! xxyy ! xxy* ! xxAA ! xxA* ! xyyy ! xyy* ! xAAA ! xAA* ! yyyA ! yyy* ! yyAA ! yyA* ! yAAA ! yAA* ! ??** ! ?*** ! ****
 * +Derivations for low hand shape xxy* (a low pair, a low card and a high card) on the turn
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   3,840 || 0.0197348 ||align=right|     49.7 : 1
 * align=right|   8,960 || 0.0460479 ||align=right|     20.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   5,760 || 0.0296022 ||align=right|     32.8 : 1
 * align=right|  10,080 || 0.0518039 ||align=right|     18.3 : 1
 * $$\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$$
 * align=right|  24,320 || 0.1249872 ||align=right|      7.0 : 1
 * align=right|  33,600 || 0.1726796 ||align=right|      4.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   2,560 || 0.0131565 ||align=right|     75.0 : 1
 * align=right|   4,480 || 0.0230239 ||align=right|     42.4 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   3,840 || 0.0197348 ||align=right|     49.7 : 1
 * align=right|   6,720 || 0.0345359 ||align=right|     28.0 : 1
 * colspan=2| Total
 * align=right|  40,320 || 0.2072155 ||align=right|      3.8 : 1
 * align=right|  63,840 || 0.3280913 ||align=right|      2.0 : 1
 * align=right|  63,840 || 0.3280913 ||align=right|      2.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      72 || 0.0003700 ||align=right|  2,701.5 : 1
 * align=right|      84 || 0.0004317 ||align=right|  2,315.4 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     240 || 0.0012334 ||align=right|    809.8 : 1
 * align=right|     336 || 0.0017268 ||align=right|    578.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     144 || 0.0007401 ||align=right|  1,350.3 : 1
 * align=right|     168 || 0.0008634 ||align=right|  1,157.2 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     216 || 0.0124884 ||align=right|     79.1 : 1
 * align=right|     252 || 0.0145698 ||align=right|     67.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,440 || 0.0074006 ||align=right|    134.1 : 1
 * align=right|   2,016 || 0.0103608 ||align=right|     95.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,440 || 0.0074006 ||align=right|    134.1 : 1
 * align=right|   2,016 || 0.0103608 ||align=right|     95.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|   2,736 || 0.0140611 ||align=right|     70.1 : 1
 * align=right|   2,520 || 0.0129510 ||align=right|     76.2 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|   9,120 || 0.0468702 ||align=right|     20.3 : 1
 * align=right|  10,080 || 0.0518039 ||align=right|     18.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     720 || 0.0037003 ||align=right|    269.3 : 1
 * align=right|   1,008 || 0.0051804 ||align=right|    192.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,160 || 0.0111008 ||align=right|     89.1 : 1
 * align=right|   3,024 || 0.0155412 ||align=right|     63.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|  13,680 || 0.0703053 ||align=right|     13.2 : 1
 * align=right|  15,120 || 0.0777058 ||align=right|     11.9 : 1
 * colspan=2| Total
 * align=right|  31,968 || 0.1642923 ||align=right|      5.1 : 1
 * align=right|  36,624 || 0.1882208 ||align=right|      4.3 : 1
 * align=right|  36,624 || 0.1882208 ||align=right|      4.3 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       6 || 0.0000308 ||align=right| 32,429.0 : 1
 * align=right|       7 || 0.0000360 ||align=right| 27,796.1 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     480 || 0.0024669 ||align=right|    404.4 : 1
 * align=right|     672 || 0.0034536 ||align=right|    288.6 : 1
 * $$\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     540 || 0.0027752 ||align=right|    359.3 : 1
 * align=right|     756 || 0.0038853 ||align=right|    256.4 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$$
 * align=right|     456 || 0.0023435 ||align=right|    425.7 : 1
 * align=right|     420 || 0.0021585 ||align=right|    462.3 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|  13,680 || 0.0703053 ||align=right|     13.2 : 1
 * align=right|  15,120 || 0.0777058 ||align=right|     11.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 2} \end{matrix}$$
 * align=right|       3 || 0.0000154 ||align=right| 64,859.0 : 1
 * align=right|       3 || 0.0000154 ||align=right| 64,859.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|      57 || 0.0002929 ||align=right|  3,412.7 : 1
 * align=right|      45 || 0.0002313 ||align=right|  4,323.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      36 || 0.0001850 ||align=right|  5,404.0 : 1
 * align=right|      42 || 0.0002158 ||align=right|  4,631.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|     456 || 0.0023435 ||align=right|    425.7 : 1
 * align=right|     420 || 0.0021585 ||align=right|    462.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 3} \end{matrix}$$
 * align=right|       2 || 0.0000103 ||align=right| 97,289.0 : 1
 * align=right|       2 || 0.0000103 ||align=right| 97,289.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|     114 || 0.0005859 ||align=right|  1,705.8 : 1
 * align=right|      90 || 0.0004625 ||align=right|  2,161.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      48 || 0.0002467 ||align=right|  4,052.8 : 1
 * align=right|      56 || 0.0002878 ||align=right|  3,473.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|   1,368 || 0.0070305 ||align=right|    141.2 : 1
 * align=right|   1,260 || 0.0064755 ||align=right|    153.4 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      24 || 0.0001233 ||align=right|  8,106.5 : 1
 * align=right|      28 || 0.0001439 ||align=right|  6,948.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{51-4r \choose 1} \end{matrix}$$
 * align=right|      19 || 0.0000976 ||align=right| 10,240.1 : 1
 * align=right|      15 || 0.0000771 ||align=right| 12,971.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     108 || 0.0005550 ||align=right|  1,800.7 : 1
 * align=right|     126 || 0.0006475 ||align=right|  1,543.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|   1,368 || 0.0070305 ||align=right|    141.2 : 1
 * align=right|   1,260 || 0.0064755 ||align=right|    153.4 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      72 || 0.0003700 ||align=right|  2,701.5 : 1
 * align=right|      84 || 0.0004317 ||align=right|  2,315.4 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|   2,052 || 0.0105458 ||align=right|     93.8 : 1
 * align=right|   1,890 || 0.0097132 ||align=right|    102.0 : 1
 * $$\begin{matrix} {4r-3 \choose 2}{51-4r \choose 2} \end{matrix}$$
 * align=right|  69,426 || 0.3567993 ||align=right|      1.8 : 1
 * align=right|  55,440 || 0.2849214 ||align=right|      2.5 : 1
 * $$\begin{matrix} {4r-3 \choose 1}{51-4r \choose 3} \end{matrix}$$
 * align=right|  28,101 || 0.1444187 ||align=right|      5.9 : 1
 * align=right|  15,015 || 0.0771662 ||align=right|     12.0 : 1
 * $$\begin{matrix} {51-4r \choose 4} \end{matrix}$$
 * align=right|   3,876 || 0.0199198 ||align=right|     49.2 : 1
 * align=right|   1,365 || 0.0070151 ||align=right|    141.5 : 1
 * colspan=2| Total
 * align=right| 122,292 || 0.6284921 ||align=right|      0.6 : 1
 * align=right|  94,116 || 0.4836879 ||align=right|      1.1 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=17| Make low ! ABCDE ! AABCD ! ABCD* ! AABBC ! AABC* ! AAABC ! ABC** ! xABCD ! xABC* ! xAABC ! xxABC ! xyABC ! yyABC ! yABCD ! yABC* ! yAABC ! rowspan=27| Miss low ! xxyyA ! xxyAB ! xxyAA ! xxyA* ! xxAAB ! xxAB* ! xyyyA ! xyyAA ! xyyAB ! xyyA* ! xyAAA ! xyAAB ! xyAA* ! xyAB* ! xyA** ! xAAAB ! xAABB ! xAAB* ! xAB** ! yyyAB ! yyAAB ! yyAB* ! yAAAB ! yAABB ! yAAB* ! yAB** ! rowspan=31| No low ! AAAAB ! AAAA* ! AAABB ! AAAB* ! AABB* ! AAA** ! AAB** ! xxyyy ! xxyy* ! xxy** ! xxAAA ! xxAA* ! xxA** ! xyyy* ! xyy** ! xAAAA ! xAAA* ! xAA** ! yyyAA ! yyyA* ! yyy** ! yyAAA ! yyAA* ! yyA** ! yAAAA ! yAAA* ! yAA** ! ??*** ! ?**** ! *****
 * +Derivations for low hand shape xxy* (a low pair, a low card and a high card) on the river
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-2 \choose 5}{4 \choose 1}^5 \end{matrix}$$
 * align=right|   6,144 || 0.0035881 ||align=right|    277.7 : 1
 * align=right|  21,504 || 0.0125585 ||align=right|     78.6 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * align=right|  53,760 || 0.0313963 ||align=right|     30.9 : 1
 * $$\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4{51-4r \choose 1} \end{matrix}$$
 * align=right|  72,960 || 0.0426093 ||align=right|     22.5 : 1
 * align=right| 134,400 || 0.0784907 ||align=right|     11.7 : 1
 * $$\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   8,640 || 0.0050458 ||align=right|    197.2 : 1
 * align=right|  15,120 || 0.0088302 ||align=right|    112.2 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right| 109,440 || 0.0639139 ||align=right|     14.6 : 1
 * align=right| 151,200 || 0.0883021 ||align=right|     10.3 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * align=right|   6,720 || 0.0039245 ||align=right|    253.8 : 1
 * $$\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{51-4r \choose 2} \end{matrix}$$
 * align=right| 218,880 || 0.1278278 ||align=right|      6.8 : 1
 * align=right| 235,200 || 0.1373588 ||align=right|      6.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   7,680 || 0.0044852 ||align=right|    222.0 : 1
 * align=right|  17,920 || 0.0104654 ||align=right|     94.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$$
 * align=right|  48,640 || 0.0284062 ||align=right|     34.2 : 1
 * align=right|  67,200 || 0.0392454 ||align=right|     24.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  11,520 || 0.0067278 ||align=right|    147.6 : 1
 * align=right|  20,160 || 0.0117736 ||align=right|     83.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   1,280 || 0.0007475 ||align=right|  1,336.7 : 1
 * align=right|   2,240 || 0.0013082 ||align=right|    763.4 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   7,680 || 0.0044852 ||align=right|    222.0 : 1
 * align=right|  13,440 || 0.0078491 ||align=right|    126.4 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * align=right|   6,720 || 0.0039245 ||align=right|    253.8 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|  11,520 || 0.0067278 ||align=right|    147.6 : 1
 * align=right|  26,880 || 0.0156981 ||align=right|     62.7 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$$
 * align=right|  72,960 || 0.0426093 ||align=right|     22.5 : 1
 * align=right| 100,800 || 0.0588681 ||align=right|     16.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  17,280 || 0.0100917 ||align=right|     98.1 : 1
 * align=right|  30,240 || 0.0176604 ||align=right|     55.6 : 1
 * colspan=2| Total
 * align=right| 625,344 || 0.3652062 ||align=right|      1.7 : 1
 * align=right| 903,504 || 0.5276540 ||align=right|      0.9 : 1
 * align=right| 903,504 || 0.5276540 ||align=right|      0.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      72 || 0.0000420 ||align=right| 23,781.0 : 1
 * align=right|      84 || 0.0000491 ||align=right| 20,383.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     108 || 0.0000631 ||align=right| 15,853.7 : 1
 * align=right|     126 || 0.0000736 ||align=right| 13,588.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{50-4r \choose 1} \end{matrix}$$
 * align=right|   1,368 || 0.0007989 ||align=right|  1,250.7 : 1
 * align=right|   1,260 || 0.0007359 ||align=right|  1,358.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|   4,560 || 0.0026631 ||align=right|    374.5  : 1
 * align=right|   5,040 || 0.0029434 ||align=right|    338.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      48 || 0.0000280 ||align=right| 35,672.0 : 1
 * align=right|      56 || 0.0000327 ||align=right| 30,575.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     216 || 0.0001261 ||align=right|  7,926.3 : 1
 * align=right|     252 || 0.0001472 ||align=right|  6,793.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,440 || 0.0008410 ||align=right|  1,188.1 : 1
 * align=right|   2,016 || 0.0011774 ||align=right|    848.4 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|   2,736 || 0.0015978 ||align=right|    624.8 : 1
 * align=right|   2,520 || 0.0014717 ||align=right|    678.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     144 || 0.0000841 ||align=right| 11,890.0 : 1
 * align=right|     168 || 0.0000981 ||align=right| 10,191.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * align=right|   6,048 || 0.0035321 ||align=right|    282.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|   4,104 || 0.0023968 ||align=right|    416.2 : 1
 * align=right|   3,780 || 0.0022076 ||align=right|    452.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|  27,360 || 0.0159785 ||align=right|     61.6 : 1
 * align=right|  30,240 || 0.0176604 ||align=right|     55.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$$
 * align=right|  24,624 || 0.0143806 ||align=right|     68.5 : 1
 * align=right|  17,640 || 0.0103019 ||align=right|     96.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     960 || 0.0005606 ||align=right|  1,782.7 : 1
 * align=right|   1,344 || 0.0007849 ||align=right|  1,273.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|   1,080 || 0.0006307 ||align=right|  1,584.5 : 1
 * align=right|   1,512 || 0.0008830 ||align=right|  1,131.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|  27,360 || 0.0159785 ||align=right|     61.6 : 1
 * align=right|  30,240 || 0.0176604 ||align=right|     55.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{51-4r \choose 2} \end{matrix}$$
 * align=right|  82,080 || 0.0479354 ||align=right|     19.9 : 1
 * align=right|  70,560 || 0.0412076 ||align=right|     23.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     240 || 0.0001402 ||align=right|  7,133.6 : 1
 * align=right|     336 || 0.0001962 ||align=right|  5,095.1 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,160 || 0.0012615 ||align=right|    791.7 : 1
 * align=right|   3,024 || 0.0017660 ||align=right|    565.2 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|  13,680 || 0.0079892 ||align=right|    124.2 : 1
 * align=right|  15,120 || 0.0088302 ||align=right|    112.2 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,440 || 0.0008410 ||align=right|  1,188.1 : 1
 * align=right|   2,016 || 0.0011774 ||align=right|    848.4 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|   1,620 || 0.0009461 ||align=right|  1,056.0 : 1
 * align=right|   2,268 || 0.0013245 ||align=right|    754.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|  41,040 || 0.0239677 ||align=right|     40.7 : 1
 * align=right|  45,360 || 0.0264906 ||align=right|     36.7 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{51-4r \choose 2} \end{matrix}$$
 * align=right| 123,120 || 0.0719031 ||align=right|     12.9 : 1
 * align=right| 105,840 || 0.0618115 ||align=right|     15.2 : 1
 * colspan=2| Total
 * align=right| 367,320 || 0.2145180 ||align=right|      3.7 : 1
 * align=right| 348,866 || 0.2037407 ||align=right|      3.9 : 1
 * align=right| 348,866 || 0.2037407 ||align=right|      3.9 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 4}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     120 || 0.0000701 ||align=right| 15,019.2 : 1
 * align=right|     168 || 0.0000613 ||align=right| 16,306.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 4}{51-4r \choose 1} \end{matrix}$$
 * align=right|     114 || 0.0000666 ||align=right| 15,853.7 : 1
 * align=right|     105 || 0.0000572 ||align=right| 17,471.5 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|  10,080 || 0.0058868 ||align=right|    168.9 : 1
 * align=right|   9,120 || 0.0053262 ||align=right|    186.8 : 1
 * $$\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|  11,340 || 0.0066227 ||align=right|    150.0 : 1
 * align=right|  10,260 || 0.0059919 ||align=right|    165.9 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{51-4r \choose 2} \end{matrix}$$
 * align=right|   2,940 || 0.0017170 ||align=right|    581.4 : 1
 * align=right|   4,104 || 0.0023968 ||align=right|    416.2 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$$
 * align=right| 105,840 || 0.0618115 ||align=right|     15.2 : 1
 * align=right| 123,120 || 0.0719031 ||align=right|     12.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 3} \end{matrix}$$
 * align=right|       1 || 0.0000006 ||align=right|1,712,303.0 : 1
 * align=right|       1 || 0.0000006 ||align=right|1,712,303.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|      57 || 0.0000333 ||align=right| 30,039.4 : 1
 * align=right|      45 || 0.0000263 ||align=right| 38,050.2 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{1 \choose 1}{3 \choose 1}{51-4r \choose 2} \end{matrix}$$
 * align=right|     513 || 0.0002996 ||align=right|  3,336.8 : 1
 * align=right|     315 || 0.0001840 ||align=right|  5,434.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      24 || 0.0000140 ||align=right| 71,345.0 : 1
 * align=right|      28 || 0.0000164 ||align=right| 61,152.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|     684 || 0.0003995 ||align=right|  2,502.4 : 1
 * align=right|     630 || 0.0003679 ||align=right|  2,716.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$$
 * align=right|   4,104 || 0.0023968 ||align=right|    416.2 : 1
 * align=right|   2,940 || 0.0017170 ||align=right|    581.4 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 3}{51-4r \choose 1} \end{matrix}$$
 * align=right|      38 || 0.0000222 ||align=right| 45,059.6 : 1
 * align=right|      30 || 0.0000175 ||align=right| 57,075.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{1 \choose 1}{3 \choose 2}{51-4r \choose 2} \end{matrix}$$
 * align=right|   1,026 || 0.0005992 ||align=right|  1,667.9 : 1
 * align=right|     630 || 0.0003679 ||align=right|  2,716.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|      12 || 0.0000070 ||align=right|142,691.0 : 1
 * align=right|      14 || 0.0000082 ||align=right|122,306.4 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$$
 * align=right|     912 || 0.0005326 ||align=right|  1,876.5 : 1
 * align=right|     840 || 0.0004906 ||align=right|  2,037.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 2} \end{matrix}$$
 * align=right|  12,312 || 0.0071903 ||align=right|    138.1 : 1
 * align=right|   8,820 || 0.0051510 ||align=right|    193.1 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      36 || 0.0000210 ||align=right| 47,563.0 : 1
 * align=right|      42 || 0.0000245 ||align=right| 40,768.1 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|     456 || 0.0002663 ||align=right|  3,754.1 : 1
 * align=right|     420 || 0.0002453 ||align=right|  4,075.9 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{51-4r \choose 2} \end{matrix}$$
 * align=right|     171 || 0.0000999 ||align=right| 10,012.5 : 1
 * align=right|     105 || 0.0000613 ||align=right| 16,306.7 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      72 || 0.0000420 ||align=right| 23,781.0 : 1
 * align=right|      84 || 0.0000491 ||align=right| 20,383.6 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|   2,052 || 0.0011984 ||align=right|    833.5 : 1
 * align=right|   1,890 || 0.0011038 ||align=right|    905.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$$
 * align=right|  12,312 || 0.0071903 ||align=right|    138.1 : 1
 * align=right|   8,820 || 0.0051510 ||align=right|    193.1 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|      18 || 0.0000105 ||align=right| 95,127.0 : 1
 * align=right|      21 || 0.0000123 ||align=right| 81,537.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$$
 * align=right|   1,368 || 0.0007989 ||align=right|  1,250.7 : 1
 * align=right|   1,260 || 0.0007359 ||align=right|  1,358.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{51-4r \choose 2} \end{matrix}$$
 * align=right|  18,468 || 0.0107855 ||align=right|     91.7 : 1
 * align=right|  13,230 || 0.0077264 ||align=right|    128.4 : 1
 * $$\begin{matrix} {4r-3 \choose 2}{51-4r \choose 3} \end{matrix}$$
 * align=right| 393,414 || 0.2297571 ||align=right|      3.4 : 1
 * align=right| 240,240 || 0.1403022 ||align=right|      6.1 : 1
 * $$\begin{matrix} {4r-3 \choose 1}{51-4r \choose 4} \end{matrix}$$
 * align=right| 112,404 || 0.0656449 ||align=right|     14.2 : 1
 * align=right|  45,045 || 0.0263067 ||align=right|     37.0 : 1
 * $$\begin{matrix} {51-4r \choose 5} \end{matrix}$$
 * align=right|  11,628 || 0.0067909 ||align=right|    146.3 : 1
 * align=right|   3,003 || 0.0017538 ||align=right|    569.2 : 1
 * colspan=2| Total
 * align=right| 719,640 || 0.4202758 ||align=right|      1.4 : 1
 * align=right| 459,934 || 0.2686053 ||align=right|      2.7 : 1
 * }
 * }

Derivations for making low hands with a low three of a kind and a low card
Starting hands with a low three of a kind and a low card (xxxy) can make a low hand when at least three other low ranks beside x and y appear on the board. With no high cards in the hand, there are $$52 - 4r - 0 = 52 - 4r$$ high cards and $$4r - 4 + 0 = 4r - 4$$ low cards left in the deck. The following tables show the derivations when holding a low three of a kind and a low card.


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=1| Make low ! ABC ! rowspan=4| Miss low ! xAB ! yAB ! xyA ! rowspan=11| No low ! AAA ! AAB ! xyy ! xAA ! yyy ! yyA ! yAA ! ??* ! ?** ! ***
 * +Derivations for low hand shape xxxy (a low three of a kind and a low card) on the flop
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   1,280 || 0.0740056 ||align=right|     12.5 : 1
 * align=right|   2,240 || 0.1295097 ||align=right|      6.7 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     240 || 0.0138760 ||align=right|     71.1 : 1
 * align=right|     336 || 0.0194265 ||align=right|     50.5 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     720 || 0.0416281 ||align=right|     23.0 : 1
 * align=right|   1,008 || 0.0582794 ||align=right|     16.2 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      72 || 0.0041628 ||align=right|    239.2 : 1
 * align=right|      84 || 0.0048566 ||align=right|    204.9 : 1
 * colspan=2| Total
 * align=right|   1,032 || 0.0596670 ||align=right|     15.8 : 1
 * align=right|   1,428 || 0.0825624 ||align=right|     11.1 : 1
 * align=right|   1,428 || 0.0825624 ||align=right|     11.1 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      24 || 0.0013876 ||align=right|    719.7 : 1
 * align=right|      28 || 0.0016189 ||align=right|    616.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     720 || 0.0416281 ||align=right|     23.0 : 1
 * align=right|   1,008 || 0.0582794 ||align=right|     16.2 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2} \end{matrix}$$
 * align=right|       3 || 0.0001735 ||align=right|  5,764.3 : 1
 * align=right|       3 || 0.0001735 ||align=right|  5,764.3 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      36 || 0.0020814 ||align=right|    479.4 : 1
 * align=right|      42 || 0.0024283 ||align=right|    410.8 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3} \end{matrix}$$
 * align=right|       1 || 0.0000578 ||align=right| 17,295.0 : 1
 * align=right|       1 || 0.0000578 ||align=right| 17,295.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     108 || 0.0062442 ||align=right|    159.1 : 1
 * align=right|     126 || 0.0072849 ||align=right|    136.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     216 || 0.0124884 ||align=right|     79.1 : 1
 * align=right|     252 || 0.0145698 ||align=right|     67.6 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   7,560 || 0.4370953 ||align=right|      1.3 : 1
 * align=right|   7,936 || 0.4588344 ||align=right|      1.2 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|   5,320 || 0.3075856 ||align=right|      2.3 : 1
 * align=right|   3,840 || 0.2220167 ||align=right|      3.5 : 1
 * $$\begin{matrix} {52-4r \choose 3} \end{matrix}$$
 * align=right|   1,140 || 0.0659112 ||align=right|     14.2 : 1
 * align=right|     560 || 0.0323774 ||align=right|     29.9 : 1
 * colspan=2| Total
 * align=right|  14,984 || 0.8663275 ||align=right|      0.2 : 1
 * align=right|  13,628 || 0.7879278 ||align=right|      0.3 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=6| Make low ! ABCD ! AABC ! ABC* ! xABC ! yABC ! rowspan=10| Miss low ! xyyA ! xyAA ! xAAB ! xyAB ! xyA* ! xAB* ! yyAB ! yAAB ! yAB* ! rowspan=19| No low ! AAAA ! AAAB ! AABB ! AAA* ! AAB* ! xyyy ! xyy* ! xAAA ! xAA* ! yyyA ! yyy* ! yyAA ! yyA* ! yAAA ! yAA* ! ??** ! ?*** ! ****
 * +Derivations for low hand shape xxxy (a low three of a kind and a low card) on the turn
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   8,960 || 0.0460479 ||align=right|     20.7 : 1
 * align=right|  17,920 || 0.0920958 ||align=right|      9.9 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  10,080 || 0.0518039 ||align=right|     18.3 : 1
 * align=right|  16,128 || 0.0828862 ||align=right|     11.1 : 1
 * $$\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$$
 * align=right|  25,600 || 0.1315654 ||align=right|      6.6 : 1
 * align=right|  35,840 || 0.1841916 ||align=right|      4.4 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   1,280 || 0.0065783 ||align=right|    151.0 : 1
 * align=right|   2,240 || 0.0115120 ||align=right|     85.9 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   3,840 || 0.0197348 ||align=right|     49.7 : 1
 * align=right|   6,720 || 0.0345359 ||align=right|     28.0 : 1
 * colspan=2| Total
 * align=right|  40,320 || 0.2072155 ||align=right|      3.8 : 1
 * align=right|  63,840 || 0.3280913 ||align=right|      2.0 : 1
 * align=right|  63,840 || 0.3280913 ||align=right|      2.0 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      72 || 0.0003700 ||align=right|  2,701.5 : 1
 * align=right|      84 || 0.0004317 ||align=right|  2,315.4 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     108 || 0.0005550 ||align=right|  1,800.7 : 1
 * align=right|     126 || 0.0006475 ||align=right|  1,543.3 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     720 || 0.0037003 ||align=right|    269.3 : 1
 * align=right|   1,008 || 0.0051804 ||align=right|    192.0 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     720 || 0.0037003 ||align=right|    269.3 : 1
 * align=right|   1,008 || 0.0051804 ||align=right|    192.0 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   1,440 || 0.0074006 ||align=right|    134.1 : 1
 * align=right|   1,344 || 0.0069072 ||align=right|    143.8 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|   4,800 || 0.0246685 ||align=right|     39.5 : 1
 * align=right|   5,376 || 0.0276287 ||align=right|     35.2 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     720 || 0.0037003 ||align=right|    269.3 : 1
 * align=right|   1,008 || 0.0051804 ||align=right|    192.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,160 || 0.0111008 ||align=right|     89.1 : 1
 * align=right|   3,024 || 0.0155412 ||align=right|     63.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  14,400 || 0.0740056 ||align=right|     12.5 : 1
 * align=right|  16,128 || 0.0828862 ||align=right|     11.1 : 1
 * colspan=2| Total
 * align=right|  25,140 || 0.1292014 ||align=right|      6.7 : 1
 * align=right|  29,106 || 0.1495837 ||align=right|      5.7 : 1
 * align=right|  29,106 || 0.1495837 ||align=right|      5.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       6 || 0.0000308 ||align=right| 32,429.0 : 1
 * align=right|       7 || 0.0000360 ||align=right| 27,796.1 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     480 || 0.0024669 ||align=right|    404.4 : 1
 * align=right|     672 || 0.0034536 ||align=right|    288.6 : 1
 * $$\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     540 || 0.0027752 ||align=right|    359.3 : 1
 * align=right|     756 || 0.0038853 ||align=right|    256.4 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|     480 || 0.0024669 ||align=right|    404.4 : 1
 * align=right|     448 || 0.0023024 ||align=right|    433.3 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|  14,400 || 0.0740056 ||align=right|     12.5 : 1
 * align=right|  16,128 || 0.0828862 ||align=right|     11.1 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 3} \end{matrix}$$
 * align=right|       1 || 0.0000051 ||align=right|194,579.0 : 1
 * align=right|       1 || 0.0000051 ||align=right|194,579.0 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|      60 || 0.0003084 ||align=right|  3,242.0 : 1
 * align=right|      48 || 0.0002467 ||align=right|  4,052.8 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      24 || 0.0001233 ||align=right|  8,106.5 : 1
 * align=right|      28 || 0.0001439 ||align=right|  6,948.3 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|     720 || 0.0037003 ||align=right|    269.3 : 1
 * align=right|     672 || 0.0034536 ||align=right|    288.6 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      24 || 0.0001233 ||align=right|  8,106.5 : 1
 * align=right|      28 || 0.0001439 ||align=right|  6,948.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|      20 || 0.0001028 ||align=right|  9,728.0 : 1
 * align=right|      16 || 0.0000822 ||align=right| 12,160.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     108 || 0.0005550 ||align=right|  1,800.7 : 1
 * align=right|     126 || 0.0006475 ||align=right|  1,543.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   1,440 || 0.0074006 ||align=right|    134.1 : 1
 * align=right|   1,344 || 0.0069072 ||align=right|    143.8 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      72 || 0.0003700 ||align=right|  2,701.5 : 1
 * align=right|      84 || 0.0004317 ||align=right|  2,315.4 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   2,160 || 0.0111008 ||align=right|     89.1 : 1
 * align=right|   2,016 || 0.0103608 ||align=right|     95.5 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 2} \end{matrix}$$
 * align=right|  71,820 || 0.3691027 ||align=right|      1.7 : 1
 * align=right|  59,520 || 0.3058896 ||align=right|      2.3 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 3} \end{matrix}$$
 * align=right|  31,920 || 0.1640456 ||align=right|      5.1 : 1
 * align=right|  17,920 || 0.0920958 ||align=right|      9.9 : 1
 * $$\begin{matrix} {52-4r \choose 4} \end{matrix}$$
 * align=right|   4,845 || 0.0248998 ||align=right|     39.2 : 1
 * align=right|   1,820 || 0.0093535 ||align=right|    105.9 : 1
 * colspan=2| Total
 * align=right| 129,120 || 0.6635831 ||align=right|      0.5 : 1
 * align=right| 101,634 || 0.5223250 ||align=right|      0.9 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=16| Make low ! ABCDE ! AABCD ! ABCD* ! AABBC ! AABC* ! AAABC ! ABC** ! xABCD ! xABC* ! xAABC ! xyABC ! yyABC ! yABCD ! yABC* ! yAABC ! rowspan=21| Miss low ! xyyyA ! xyyAA ! xyyAB ! xyyA* ! xyAAA ! xyAAB ! xyAA* ! xyAB* ! xyA** ! xAAAB ! xAABB ! xAAB* ! xAB** ! yyyAB ! yyAAB ! yyAB* ! yAAAB ! yAABB ! yAAB* ! yAB** ! rowspan=25| No low ! AAAAB ! AAAA* ! AAABB ! AAAB* ! AABB* ! AAA** ! AAB** ! xyyy* ! xyy** ! xAAAA ! xAAA* ! xAA** ! yyyAA ! yyyA* ! yyy** ! yyAAA ! yyAA* ! yyA** ! yAAAA ! yAAA* ! yAA** ! ??*** ! ?**** ! *****
 * +Derivations for low hand shape xxxy (a low three of a kind and a low card) on the river
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-2 \choose 5}{4 \choose 1}^5 \end{matrix}$$
 * align=right|   6,144 || 0.0035881 ||align=right|    277.7 : 1
 * align=right|  21,504 || 0.0125585 ||align=right|     78.6 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * align=right|  53,760 || 0.0313963 ||align=right|     30.9 : 1
 * $$\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4{52-4r \choose 1} \end{matrix}$$
 * align=right|  76,800 || 0.0448518 ||align=right|     21.3 : 1
 * align=right| 143,360 || 0.0837235 ||align=right|     10.9 : 1
 * $$\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   8,640 || 0.0050458 ||align=right|    197.2 : 1
 * align=right|  15,120 || 0.0088302 ||align=right|    112.2 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right| 115,200 || 0.0672778 ||align=right|     13.9 : 1
 * align=right| 161,280 || 0.0941889 ||align=right|      9.6 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * align=right|   6,720 || 0.0039245 ||align=right|    253.8 : 1
 * $$\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 2} \end{matrix}$$
 * align=right| 243,200 || 0.1420309 ||align=right|      6.0 : 1
 * align=right| 268,800 || 0.1569815 ||align=right|      5.4 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * align=right|   8,960 || 0.0052327 ||align=right|    190.1 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$$
 * align=right|  25,600 || 0.0149506 ||align=right|     65.9 : 1
 * align=right|  35,840 || 0.0209309 ||align=right|     46.8 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   5,760 || 0.0033639 ||align=right|    296.3 : 1
 * align=right|  10,080 || 0.0058868 ||align=right|    168.9 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * align=right|   6,720 || 0.0039245 ||align=right|    253.8 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * align=right|   6,720 || 0.0039245 ||align=right|    253.8 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|  11,520 || 0.0067278 ||align=right|    147.6 : 1
 * align=right|  26,880 || 0.0156981 ||align=right|     62.7 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$$
 * align=right|  76,800 || 0.0448518 ||align=right|     21.3 : 1
 * align=right| 107,520 || 0.0627926 ||align=right|     14.9 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  17,280 || 0.0100917 ||align=right|     98.1 : 1
 * align=right|  30,240 || 0.0176604 ||align=right|     55.6 : 1
 * colspan=2| Total
 * align=right| 625,344 || 0.3652062 ||align=right|      1.7 : 1
 * align=right| 903,504 || 0.5276540 ||align=right|      0.9 : 1
 * align=right| 903,504 || 0.5276540 ||align=right|      0.9 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      24 || 0.0000140 ||align=right| 71,345.0 : 1
 * align=right|      28 || 0.0000164 ||align=right| 61,152.7 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     108 || 0.0000631 ||align=right| 15,853.7 : 1
 * align=right|     126 || 0.0000736 ||align=right| 13,588.7 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   1,440 || 0.0008410 ||align=right|  1,188.1 : 1
 * align=right|   1,344 || 0.0007849 ||align=right|  1,273.0 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      72 || 0.0000420 ||align=right| 23,781.0 : 1
 * align=right|      84 || 0.0000491 ||align=right| 20,383.6 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,160 || 0.0012615 ||align=right|    791.7 : 1
 * align=right|   3,024 || 0.0017660 ||align=right|    565.2 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   2,160 || 0.0012615 ||align=right|    791.7 : 1
 * align=right|   2,016 || 0.0011774 ||align=right|    848.4 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  14,400 || 0.0084097 ||align=right|    117.9 : 1
 * align=right|  16,128 || 0.0094189 ||align=right|    105.2 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|  13,680 || 0.0079892 ||align=right|    124.2 : 1
 * align=right|  10,080 || 0.0058868 ||align=right|    168.9 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     480 || 0.0002803 ||align=right|  3,566.3 : 1
 * align=right|     672 || 0.0003925 ||align=right|  2,547.1 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     540 || 0.0003154 ||align=right|  3,169.9 : 1
 * align=right|     756 || 0.0004415 ||align=right|  2,264.0 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|  14,400 || 0.0084097 ||align=right|    117.9 : 1
 * align=right|  16,128 || 0.0094189 ||align=right|    105.2 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{52-4r \choose 2} \end{matrix}$$
 * align=right|  45,600 || 0.0266308 ||align=right|     36.6 : 1
 * align=right|  40,320 || 0.0235472 ||align=right|     41.5 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     240 || 0.0001402 ||align=right|  7,133.6 : 1
 * align=right|     336 || 0.0001962 ||align=right|  5,095.1 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,160 || 0.0012615 ||align=right|    791.7 : 1
 * align=right|   3,024 || 0.0017660 ||align=right|    565.2 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  14,400 || 0.0084097 ||align=right|    117.9 : 1
 * align=right|  16,128 || 0.0094189 ||align=right|    105.2 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,440 || 0.0008410 ||align=right|  1,188.1 : 1
 * align=right|   2,016 || 0.0011774 ||align=right|    848.4 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|   1,620 || 0.0009461 ||align=right|  1,056.0 : 1
 * align=right|   2,268 || 0.0013245 ||align=right|    754.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|  43,200 || 0.0252292 ||align=right|     38.6 : 1
 * align=right|  48,384 || 0.0282567 ||align=right|     34.4 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{52-4r \choose 2} \end{matrix}$$
 * align=right| 136,800 || 0.0798924 ||align=right|     11.5 : 1
 * align=right| 120,960 || 0.0706417 ||align=right|     13.2 : 1
 * colspan=2| Total
 * align=right| 295,644 || 0.1726586 ||align=right|      4.8 : 1
 * align=right| 284,830 || 0.1663431 ||align=right|      5.0 : 1
 * align=right| 284,830 || 0.1663431 ||align=right|      5.0 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 4}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     120 || 0.0000701 ||align=right| 15,019.2 : 1
 * align=right|     168 || 0.0000613 ||align=right| 16,306.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 4}{52-4r \choose 1} \end{matrix}$$
 * align=right|     120 || 0.0000701 ||align=right| 14,268.2 : 1
 * align=right|     112 || 0.0000654 ||align=right| 15,287.4 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   9,600 || 0.0056065 ||align=right|    177.4 : 1
 * align=right|  10,752 || 0.0062793 ||align=right|    158.3 : 1
 * $$\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  10,800 || 0.0063073 ||align=right|    157.5 : 1
 * align=right|  12,096 || 0.0070642 ||align=right|    140.6 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{52-4r \choose 2} \end{matrix}$$
 * align=right|   4,560 || 0.0026631 ||align=right|    374.5 : 1
 * align=right|   3,360 || 0.0019623 ||align=right|    508.6 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right| 136,800 || 0.0798924 ||align=right|     11.5 : 1
 * align=right| 120,960 || 0.0706417 ||align=right|     13.2 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|      20 || 0.0000117 ||align=right| 85,614.2 : 1
 * align=right|      16 || 0.0000093 ||align=right|107,018.0 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{1 \choose 1}{3 \choose 2}{52-4r \choose 2} \end{matrix}$$
 * align=right|     570 || 0.0003329 ||align=right|  3,003.0 : 1
 * align=right|     360 || 0.0002102 ||align=right|  4,755.4 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       6 || 0.0000035 ||align=right|285,383.0 : 1
 * align=right|       7 || 0.0000041 ||align=right|244,613.9 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|     480 || 0.0002803 ||align=right|  3,566.3 : 1
 * align=right|     448 || 0.0002616 ||align=right|  3,821.1 : 1
 * $$\begin{matrix} {1 \choose 1}{1 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 2} \end{matrix}$$
 * align=right|   6,840 || 0.0039946 ||align=right|    249.3 : 1
 * align=right|   5,040 || 0.0029434 ||align=right|    338.7 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      36 || 0.0000210 ||align=right| 47,563.0 : 1
 * align=right|      42 || 0.0000245 ||align=right| 40,768.1 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|     480 || 0.0002803 ||align=right|  3,566.3 : 1
 * align=right|     448 || 0.0002616 ||align=right|  3,821.1 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 3}{52-4r \choose 2} \end{matrix}$$
 * align=right|     190 || 0.0001110 ||align=right|  9,011.1 : 1
 * align=right|     120 || 0.0000701 ||align=right| 14,268.2 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      72 || 0.0000420 ||align=right| 23,781.0 : 1
 * align=right|      84 || 0.0000491 ||align=right| 20,383.6 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   2,160 || 0.0012615 ||align=right|    791.7 : 1
 * align=right|   2,016 || 0.0011774 ||align=right|    848.4 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|  13,680 || 0.0079892 ||align=right|    124.2 : 1
 * align=right|  10,080 || 0.0058868 ||align=right|    168.9 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|      18 || 0.0000105 ||align=right| 95,127.0 : 1
 * align=right|      21 || 0.0000123 ||align=right| 81,537.3 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|   1,440 || 0.0008410 ||align=right|  1,188.1 : 1
 * align=right|   1,344 || 0.0007849 ||align=right|  1,273.0 : 1
 * $$\begin{matrix} {1 \choose 1}{3 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 2} \end{matrix}$$
 * align=right|  20,520 || 0.0119839 ||align=right|     82.4 : 1
 * align=right|  15,120 || 0.0088302 ||align=right|    112.2 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 3} \end{matrix}$$
 * align=right| 430,920 || 0.2516609 ||align=right|      3.0 : 1
 * align=right| 277,760 || 0.1622142 ||align=right|      5.2 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 4} \end{matrix}$$
 * align=right| 135,660 || 0.0792266 ||align=right|     11.6 : 1
 * align=right|  58,240 || 0.0340127 ||align=right|     28.4 : 1
 * $$\begin{matrix} {52-4r \choose 5} \end{matrix}$$
 * align=right|  15,504 || 0.0090545 ||align=right|    109.4 : 1
 * align=right|   4,368 || 0.0025509 ||align=right|    391.0 : 1
 * colspan=2| Total
 * align=right| 791,316 || 0.4621352 ||align=right|      1.2 : 1
 * align=right| 523,970 || 0.3060029 ||align=right|      2.3 : 1
 * }
 * }

Derivations for making low hands with two low pair
Starting hands with two low pair (xxyy) can make a low hand when at least three other low ranks beside x and y appear on the board. With no high cards in the hand, there are $$52 - 4r - 0 = 52 - 4r$$ high cards and $$4r - 4 + 0 = 4r - 4$$ low cards left in the deck. The following tables show the derivations when holding two low pair.


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=1| Make low ! ABC ! rowspan=3| Miss low ! xAB ! xyA ! rowspan=9| No low ! AAA ! AAB ! xxy ! xxA ! xAA ! ??* ! ?** ! ***
 * +Derivations for low hand shape xxyy (two low pair) on the flop
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   1,280 || 0.0740056 ||align=right|     12.5 : 1
 * align=right|   2,240 || 0.1295097 ||align=right|      6.7 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     960 || 0.0555042 ||align=right|     17.0 : 1
 * align=right|   1,344 || 0.0777058 ||align=right|     11.9 : 1
 * $$\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      96 || 0.0055504 ||align=right|    179.2 : 1
 * align=right|     112 || 0.0064755 ||align=right|    153.4 : 1
 * colspan=2| Total
 * align=right|   1,056 || 0.0610546 ||align=right|     15.4 : 1
 * align=right|   1,456 || 0.0841813 ||align=right|     10.9 : 1
 * align=right|   1,456 || 0.0841813 ||align=right|     10.9 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      24 || 0.0013876 ||align=right|    719.7 : 1
 * align=right|      28 || 0.0016189 ||align=right|    616.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     720 || 0.0416281 ||align=right|     23.0 : 1
 * align=right|   1,008 || 0.0582794 ||align=right|     16.2 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1} \end{matrix}$$
 * align=right|       4 || 0.0002313 ||align=right|  4,323.0 : 1
 * align=right|       4 || 0.0002313 ||align=right|  4,323.0 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      48 || 0.0027752 ||align=right|    359.3 : 1
 * align=right|      56 || 0.0032377 ||align=right|    307.9 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     144 || 0.0083256 ||align=right|    119.1 : 1
 * align=right|     168 || 0.0097132 ||align=right|    102.0 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   7,560 || 0.4370953 ||align=right|      1.3 : 1
 * align=right|   7,936 || 0.4588344 ||align=right|      1.2 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|   5,320 || 0.3075856 ||align=right|      2.3 : 1
 * align=right|   3,840 || 0.2220167 ||align=right|      3.5 : 1
 * $$\begin{matrix} {52-4r \choose 3} \end{matrix}$$
 * align=right|   1,140 || 0.0659112 ||align=right|     14.2 : 1
 * align=right|     560 || 0.0323774 ||align=right|     29.9 : 1
 * colspan=2| Total
 * align=right|  14,960 || 0.8649399 ||align=right|      0.2 : 1
 * align=right|  13,600 || 0.7863090 ||align=right|      0.3 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=5| Make low ! ABCD ! AABC ! ABC* ! xABC ! rowspan=8| Miss low ! xxyA ! xxAB ! xyAA ! xAAB ! xyAB ! xyA* ! xAB* ! rowspan=15| No low ! AAAA ! AAAB ! AABB ! AAA* ! AAB* ! xxyy ! xxy* ! xxAA ! xxA* ! xAAA ! xAA* ! ??** ! ?*** ! ****
 * +Derivations for low hand shape xxyy (two low pair) on the turn
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   3,840 || 0.0197348 ||align=right|     49.7 : 1
 * align=right|   8,960 || 0.0460479 ||align=right|     20.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   5,760 || 0.0296022 ||align=right|     32.8 : 1
 * align=right|  10,080 || 0.0518039 ||align=right|     18.3 : 1
 * $$\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$$
 * align=right|  25,600 || 0.1315654 ||align=right|      6.6 : 1
 * align=right|  35,840 || 0.1841916 ||align=right|      4.4 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   5,120 || 0.0263131 ||align=right|     37.0 : 1
 * align=right|   8,960 || 0.0460479 ||align=right|     20.7 : 1
 * colspan=2| Total
 * align=right|  40,320 || 0.2072155 ||align=right|      3.8 : 1
 * align=right|  63,840 || 0.3280913 ||align=right|      2.0 : 1
 * align=right|  63,840 || 0.3280913 ||align=right|      2.0 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      96 || 0.0004934 ||align=right|  2,025.9 : 1
 * align=right|     112 || 0.0005756 ||align=right|  1,736.3 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     480 || 0.0024669 ||align=right|    404.4 : 1
 * align=right|     672 || 0.0034536 ||align=right|    288.6 : 1
 * $$\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     144 || 0.0007401 ||align=right|  1,350.3 : 1
 * align=right|     168 || 0.0008634 ||align=right|  1,157.2 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,880 || 0.0148011 ||align=right|     66.6 : 1
 * align=right|   4,032 || 0.0207216 ||align=right|     47.3 : 1
 * $$\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     960 || 0.0049337 ||align=right|    201.7 : 1
 * align=right|   1,344 || 0.0069072 ||align=right|    143.8 : 1
 * $$\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   1,920 || 0.0098674 ||align=right|    100.3 : 1
 * align=right|   1,792 || 0.0092096 ||align=right|    107.6 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  19,200 || 0.0986741 ||align=right|      9.1 : 1
 * align=right|  21,504 || 0.1105150 ||align=right|      8.0 : 1
 * colspan=2| Total
 * align=right|  25,680 || 0.1319766 ||align=right|      6.6 : 1
 * align=right|  29,624 || 0.1522459 ||align=right|      5.6 : 1
 * align=right|  29,624 || 0.1522459 ||align=right|      5.6 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       6 || 0.0000308 ||align=right| 32,429.0 : 1
 * align=right|       7 || 0.0000360 ||align=right| 27,796.1 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     480 || 0.0024669 ||align=right|    404.4 : 1
 * align=right|     672 || 0.0034536 ||align=right|    288.6 : 1
 * $$\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     540 || 0.0027752 ||align=right|    359.3 : 1
 * align=right|     756 || 0.0038853 ||align=right|    256.4 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|     480 || 0.0024669 ||align=right|    404.4 : 1
 * align=right|     448 || 0.0023024 ||align=right|    433.3 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|  14,400 || 0.0740056 ||align=right|     12.5 : 1
 * align=right|  16,128 || 0.0828862 ||align=right|     11.1 : 1
 * $$\begin{matrix} {2 \choose 2}{2 \choose 2}^2 \end{matrix}$$
 * align=right|       1 || 0.0000051 ||align=right|194,579.0 : 1
 * align=right|       1 || 0.0000051 ||align=right|194,579.0 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|      80 || 0.0004111 ||align=right|  2,431.3 : 1
 * align=right|      64 || 0.0003289 ||align=right|  3,039.3 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      72 || 0.0003700 ||align=right|  2,701.5 : 1
 * align=right|      84 || 0.0004317 ||align=right|  2,315.4 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|     960 || 0.0049337 ||align=right|    201.7 : 1
 * align=right|     896 || 0.0046048 ||align=right|    216.2 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      96 || 0.0004934 ||align=right|  2,025.9 : 1
 * align=right|     112 || 0.0005756 ||align=right|  1,736.3 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   2,880 || 0.0148011 ||align=right|     66.6 : 1
 * align=right|   2,688 || 0.0138144 ||align=right|     71.4 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 2} \end{matrix}$$
 * align=right|  71,820 || 0.3691027 ||align=right|      1.7 : 1
 * align=right|  59,520 || 0.3058896 ||align=right|      2.3 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 3} \end{matrix}$$
 * align=right|  31,920 || 0.1640456 ||align=right|      5.1 : 1
 * align=right|  17,920 || 0.0920958 ||align=right|      9.9 : 1
 * $$\begin{matrix} {52-4r \choose 4} \end{matrix}$$
 * align=right|   4,845 || 0.0248998 ||align=right|     39.2 : 1
 * align=right|   1,820 || 0.0093535 ||align=right|    105.9 : 1
 * colspan=2| Total
 * align=right| 128,580 || 0.6608079 ||align=right|      0.5 : 1
 * align=right| 101,116 || 0.5196629 ||align=right|      0.9 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=13| Make low ! ABCDE ! AABCD ! ABCD* ! AABBC ! AABC* ! AAABC ! ABC** ! xABCD ! xABC* ! xAABC ! xxABC ! xyABC ! rowspan=16| Miss low ! xxyyA ! xxyAB ! xxyAA ! xxyA* ! xxAAB ! xxAB* ! xyAAA ! xyAAB ! xyAA* ! xyAB* ! xyA** ! xAAAB ! xAABB ! xAAB* ! xAB** ! rowspan=19| No low ! AAAAB ! AAAA* ! AAABB ! AAAB* ! AABB* ! AAA** ! AAB** ! xxyy* ! xxy** ! xxAAA ! xxAA* ! xxA** ! xAAAA ! xAAA* ! xAA** ! ??*** ! ?**** ! *****
 * +Derivations for low hand shape xxyy (two low pair) on the river
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-2 \choose 5}{4 \choose 1}^5 \end{matrix}$$
 * align=right|   6,144 || 0.0035881 ||align=right|    277.7 : 1
 * align=right|  21,504 || 0.0125585 ||align=right|     78.6 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * align=right|  53,760 || 0.0313963 ||align=right|     30.9 : 1
 * $$\begin{matrix} {r-2 \choose 4}{4 \choose 1}^4{52-4r \choose 1} \end{matrix}$$
 * align=right|  76,800 || 0.0448518 ||align=right|     21.3 : 1
 * align=right| 143,360 || 0.0837235 ||align=right|     10.9 : 1
 * $$\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   8,640 || 0.0050458 ||align=right|    197.2 : 1
 * align=right|  15,120 || 0.0088302 ||align=right|    112.2 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right| 115,200 || 0.0672778 ||align=right|     13.9 : 1
 * align=right| 161,280 || 0.0941889 ||align=right|      9.6 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * align=right|   6,720 || 0.0039245 ||align=right|    253.8 : 1
 * $$\begin{matrix} {r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 2} \end{matrix}$$
 * align=right| 243,200 || 0.1420309 ||align=right|      6.0 : 1
 * align=right| 268,800 || 0.1569815 ||align=right|      5.4 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|  15,360 || 0.0089704 ||align=right|    110.5 : 1
 * align=right|  35,840 || 0.0209309 ||align=right|     46.8 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$$
 * align=right| 102,400 || 0.0598025 ||align=right|     15.7 : 1
 * align=right| 143,360 || 0.0837235 ||align=right|     10.9 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * align=right|  40,320 || 0.0235472 ||align=right|     41.5 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   2,560 || 0.0014951 ||align=right|    667.9 : 1
 * align=right|   4,480 || 0.0026164 ||align=right|    381.2 : 1
 * $$\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   5,120 || 0.0029901 ||align=right|    333.4 : 1
 * align=right|   8,960 || 0.0052327 ||align=right|    190.1 : 1
 * colspan=2| Total
 * align=right| 625,344 || 0.3652062 ||align=right|      1.7 : 1
 * align=right| 903,504 || 0.5276540 ||align=right|      0.9 : 1
 * align=right| 903,504 || 0.5276540 ||align=right|      0.9 : 1
 * $$\begin{matrix} {2 \choose 2}{2 \choose 2}^2{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      24 || 0.0000140 ||align=right| 71,345.0 : 1
 * align=right|      28 || 0.0000164 ||align=right| 61,152.7 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{r-2 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     960 || 0.0005606 ||align=right|  1,782.7 : 1
 * align=right|   1,344 || 0.0007849 ||align=right|  1,273.0 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     144 || 0.0000841 ||align=right| 11,890.0 : 1
 * align=right|     168 || 0.0000981 ||align=right| 10,191.3 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   1,920 || 0.0011213 ||align=right|    890.8 : 1
 * align=right|   1,792 || 0.0010465 ||align=right|    954.5 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,440 || 0.0008410 ||align=right|  1,188.1 : 1
 * align=right|   2,016 || 0.0011774 ||align=right|    848.4 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|   9,600 || 0.0056065 ||align=right|    177.4 : 1
 * align=right|  10,752 || 0.0062793 ||align=right|    158.3 : 1
 * $$\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      96 || 0.0000561 ||align=right| 17,835.5 : 1
 * align=right|     112 || 0.0000654 ||align=right| 15,287.4 : 1
 * $$\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,880 || 0.0016819 ||align=right|    593.6 : 1
 * align=right|   4,032 || 0.0023547 ||align=right|    423.7 : 1
 * $$\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   2,880 || 0.0016819 ||align=right|    593.6 : 1
 * align=right|   2,688 || 0.0015698 ||align=right|    636.0 : 1
 * $$\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  19,200 || 0.0112130 ||align=right|     88.2 : 1
 * align=right|  21,504 || 0.0125585 ||align=right|     78.6 : 1
 * $$\begin{matrix} {2 \choose 2}{2 \choose 1}^2{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|  18,240 || 0.0106523 ||align=right|     92.9 : 1
 * align=right|  13,440 || 0.0078491 ||align=right|    126.4 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-2 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,920 || 0.0011213 ||align=right|    890.8 : 1
 * align=right|   2,688 || 0.0015698 ||align=right|    636.0 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|   2,160 || 0.0012615 ||align=right|    791.7 : 1
 * align=right|   3,024 || 0.0017660 ||align=right|    565.2 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|  57,600 || 0.0336389 ||align=right|     28.7 : 1
 * align=right|  64,512 || 0.0376756 ||align=right|     25.5 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{52-4r \choose 2} \end{matrix}$$
 * align=right| 182,400 || 0.1065231 ||align=right|      8.4 : 1
 * align=right| 161,280 || 0.0941889 ||align=right|      9.6 : 1
 * colspan=2| Total
 * align=right| 301,464 || 0.1760575 ||align=right|      4.7 : 1
 * align=right| 289,380 || 0.1690004 ||align=right|      4.9 : 1
 * align=right| 289,380 || 0.1690004 ||align=right|      4.9 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 4}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     120 || 0.0000701 ||align=right| 14,268.2 : 1
 * align=right|     168 || 0.0000981 ||align=right| 10,191.3 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 4}{52-4r \choose 1} \end{matrix}$$
 * align=right|     120 || 0.0000701 ||align=right| 14,268.2 : 1
 * align=right|     112 || 0.0000654 ||align=right| 15,287.4 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * align=right|   1,008 || 0.0005887 ||align=right|  1,697.7 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   9,600 || 0.0056065 ||align=right|    177.4 : 1
 * align=right|  10,752 || 0.0062793 ||align=right|    158.3 : 1
 * $$\begin{matrix} {r-2 \choose 2}{4 \choose 2}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  10,800 || 0.0063073 ||align=right|    157.5 : 1
 * align=right|  12,096 || 0.0070642 ||align=right|    140.6 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 3}{52-4r \choose 2} \end{matrix}$$
 * align=right|   4,560 || 0.0026631 ||align=right|    374.5 : 1
 * align=right|   3,360 || 0.0019623 ||align=right|    508.6 : 1
 * $$\begin{matrix} {r-2 \choose 1}{4 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right| 136,800 || 0.0798924 ||align=right|     11.5 : 1
 * align=right| 120,960 || 0.0706417 ||align=right|     13.2 : 1
 * $$\begin{matrix} {2 \choose 2}{2 \choose 2}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|      20 || 0.0000117 ||align=right| 85,614.2 : 1
 * align=right|      16 || 0.0000093 ||align=right|107,018.0 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{1 \choose 1}{2 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|     760 || 0.0004438 ||align=right|  2,252.0 : 1
 * align=right|     480 || 0.0002803 ||align=right|  3,566.3 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      48 || 0.0000280 ||align=right| 35,672.0 : 1
 * align=right|      56 || 0.0000327 ||align=right| 30,575.9 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   1,440 || 0.0008410 ||align=right|  1,188.1 : 1
 * align=right|   1,344 || 0.0007849 ||align=right|  1,273.0 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 2}{r-2 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|   9,120 || 0.0053262 ||align=right|    186.8 : 1
 * align=right|   6,720 || 0.0039245 ||align=right|    253.8 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|      24 || 0.0000140 ||align=right| 71,345.0 : 1
 * align=right|      28 || 0.0000164 ||align=right| 61,152.7 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|   1,920 || 0.0011213 ||align=right|    890.8 : 1
 * align=right|   1,792 || 0.0010465 ||align=right|    954.5 : 1
 * $$\begin{matrix} {2 \choose 1}{2 \choose 1}{r-2 \choose 1}{4 \choose 2}{52-4r \choose 2} \end{matrix}$$
 * align=right|  27,360 || 0.0159785 ||align=right|     61.6 : 1
 * align=right|  20,160 || 0.0117736 ||align=right|     83.9 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 3} \end{matrix}$$
 * align=right| 430,920 || 0.2516609 ||align=right|      3.0 : 1
 * align=right| 277,760 || 0.1622142 ||align=right|      5.2 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 4} \end{matrix}$$
 * align=right| 135,660 || 0.0792266 ||align=right|     11.6 : 1
 * align=right|  58,240 || 0.0340127 ||align=right|     28.4 : 1
 * $$\begin{matrix} {52-4r \choose 5} \end{matrix}$$
 * align=right|  15,504 || 0.0090545 ||align=right|    109.4 : 1
 * align=right|   4,368 || 0.0025509 ||align=right|    391.0 : 1
 * colspan=2| Total
 * align=right| 785,496 || 0.4587363 ||align=right|      1.2 : 1
 * align=right| 519,420 || 0.3033457 ||align=right|      2.3 : 1
 * }
 * }

Derivations for making low hands with three low ranks and a high card
Starting hands with three low ranks and a high card (xyz*) can make a low hand when either at least three other low ranks beside x, y and z appear on the board; or when two other low ranks beside x, y and z and at least one of the low ranks x, y or z appears. With one high card in the hand, there are $$52 - 4r - 1 = 51 - 4r$$ high cards and $$4r - 4 + 1 = 4r - 3$$ low cards left in the deck. The following tables show the derivations when holding three low ranks and a high card.


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=3| Make low ! ABC ! xAB ! rowspan=3| Miss low ! xyA ! xyz ! rowspan=10| No low ! AAA ! AAB ! xxx ! xxy ! xxA ! xAA ! ??* ! ?** ! ***
 * +Derivations for low hand shape xyz* (three low ranks and a high card) on the flop
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|     640 || 0.0370028 ||align=right|     26.0 : 1
 * align=right|   1,280 || 0.0740056 ||align=right|     12.5 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,440 || 0.0832562 ||align=right|     11.0 : 1
 * align=right|   2,160 || 0.1248844 ||align=right|      7.0 : 1
 * colspan=2| Total
 * align=right|   2,080 || 0.1202590 ||align=right|      7.3 : 1
 * align=right|   3,440 || 0.1988899 ||align=right|      4.0 : 1
 * align=right|   3,440 || 0.1988899 ||align=right|      4.0 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     540 || 0.0312211 ||align=right|     31.0 : 1
 * align=right|     648 || 0.0374653 ||align=right|     25.7 : 1
 * $$\begin{matrix} {3 \choose 3}{3 \choose 1}^3 \end{matrix}$$
 * align=right|      27 || 0.0015611 ||align=right|    639.6 : 1
 * align=right|      27 || 0.0015611 ||align=right|    639.6 : 1
 * colspan=2| Total
 * align=right|     567 || 0.0327821 ||align=right|     29.5 : 1
 * align=right|     675 || 0.0390264 ||align=right|     24.6 : 1
 * align=right|     675 || 0.0390264 ||align=right|     24.6 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      20 || 0.0011563 ||align=right|    863.8 : 1
 * align=right|      24 || 0.0013876 ||align=right|    719.7 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     480 || 0.0277521 ||align=right|     35.0 : 1
 * align=right|     720 || 0.0416281 ||align=right|     23.0 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 3} \end{matrix}$$
 * align=right|       3 || 0.0001735 ||align=right|  5,764.3 : 1
 * align=right|       3 || 0.0001735 ||align=right|  5,764.3 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|      54 || 0.0031221 ||align=right|    319.3 : 1
 * align=right|      54 || 0.0031221 ||align=right|    319.3 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     180 || 0.0104070 ||align=right|     95.1 : 1
 * align=right|     216 || 0.0124884 ||align=right|     79.1 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     270 || 0.0156105 ||align=right|     63.1 : 1
 * align=right|     324 || 0.0187327 ||align=right|     52.4 : 1
 * $$\begin{matrix} {4r-3 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|   7,714 || 0.4459991 ||align=right|      1.2 : 1
 * align=right|   7,920 || 0.4579093 ||align=right|      1.2 : 1
 * $$\begin{matrix} {4r-3 \choose 1}{51-4r \choose 2} \end{matrix}$$
 * align=right|   4,959 || 0.2867137 ||align=right|      2.5 : 1
 * align=right|   3,465 || 0.2003353 ||align=right|      4.0 : 1
 * $$\begin{matrix} {51-4r \choose 3} \end{matrix}$$
 * align=right|     969 || 0.0560245 ||align=right|     16.8 : 1
 * align=right|     455 || 0.0263067 ||align=right|     37.0 : 1
 * colspan=2| Total
 * align=right|  14,649 || 0.8469588 ||align=right|      0.2 : 1
 * align=right|  13,181 || 0.7620837 ||align=right|      0.3 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=9| Make low ! ABCD ! AABC ! ABC* ! xABC ! xxAB ! xAAB ! xyAB ! xAB* ! rowspan=7| Miss low ! xxyA ! xyAA ! xyA* ! xxyz ! xyzA ! xyz* ! rowspan=18| No low ! AAAA ! AAAB ! AABB ! AAA* ! AAB* ! xxxy ! xxxA ! xxx* ! xxyy ! xxy* ! xxAA ! xxA* ! xAAA ! xAA* ! ??** ! ?*** ! ****
 * +Derivations for low hand shape xyz* (three low ranks and a high card) on the turn
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-3 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   1,280 || 0.0065783 ||align=right|    151.0 : 1
 * align=right|   3,840 || 0.0197348 ||align=right|     49.7 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   2,880 || 0.0148011 ||align=right|     66.6 : 1
 * align=right|   5,760 || 0.0296022 ||align=right|     32.8 : 1
 * $$\begin{matrix} {r-3 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$$
 * align=right|  12,160 || 0.0624936 ||align=right|     15.0 : 1
 * align=right|  19,200 || 0.0986741 ||align=right|      9.1 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   5,760 || 0.0296022 ||align=right|     32.8 : 1
 * align=right|  11,520 || 0.0592044 ||align=right|     15.9 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,440 || 0.0074006 ||align=right|    134.1 : 1
 * align=right|   2,160 || 0.0111008 ||align=right|     89.1 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   4,320 || 0.0222017 ||align=right|     44.0 : 1
 * align=right|   6,480 || 0.0333025 ||align=right|     29.0 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   4,320 || 0.0222017 ||align=right|     44.0 : 1
 * align=right|   6,480 || 0.0333025 ||align=right|     29.0 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|  27,360 || 0.1406105 ||align=right|      6.1 : 1
 * align=right|  32,400 || 0.1665125 ||align=right|      5.0 : 1
 * colspan=2| Total
 * align=right|  59,520 || 0.3058896 ||align=right|      2.3 : 1
 * align=right|  87,840 || 0.4514339 ||align=right|      1.2 : 1
 * align=right|  87,840 || 0.4514339 ||align=right|      1.2 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,080 || 0.0055504 ||align=right|    179.2 : 1
 * align=right|   1,296 || 0.0066605 ||align=right|    149.1 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     810 || 0.0041628 ||align=right|    239.2 : 1
 * align=right|     972 || 0.0049954 ||align=right|    199.2 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|  10,260 || 0.0527290 ||align=right|     18.0 : 1
 * align=right|   9,720 || 0.0499537 ||align=right|     19.0 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 2}{3 \choose 1}^2 \end{matrix}$$
 * align=right|      81 || 0.0004163 ||align=right|  2,401.2 : 1
 * align=right|      81 || 0.0004163 ||align=right|  2,401.2 : 1
 * $$\begin{matrix} {3 \choose 3}{3 \choose 1}^3{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     540 || 0.0027752 ||align=right|    359.3 : 1
 * align=right|     648 || 0.0033302 ||align=right|    299.3 : 1
 * $$\begin{matrix} {3 \choose 3}{3 \choose 1}^3{51-4r \choose 1} \end{matrix}$$
 * align=right|     513 || 0.0026364 ||align=right|    378.3 : 1
 * align=right|     405 || 0.0020814 ||align=right|    479.4 : 1
 * colspan=2| Total
 * align=right|  13,284 || 0.0682701 ||align=right|     13.6 : 1
 * align=right|  13,122 || 0.0674376 ||align=right|     13.8 : 1
 * align=right|  13,122 || 0.0674376 ||align=right|     13.8 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       5 || 0.0000257 ||align=right| 38,915.0 : 1
 * align=right|       6 || 0.0000308 ||align=right| 32,429.0 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     320 || 0.0016446 ||align=right|    607.1 : 1
 * align=right|     480 || 0.0024669 ||align=right|    404.4 : 1
 * $$\begin{matrix} {r-3 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     360 || 0.0018501 ||align=right|    539.5 : 1
 * align=right|     540 || 0.0027752 ||align=right|    359.3 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$$
 * align=right|     380 || 0.0019529 ||align=right|    511.1 : 1
 * align=right|     360 || 0.0018501 ||align=right|    539.5 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|   9,120 || 0.0468702 ||align=right|     20.3 : 1
 * align=right|  10,800 || 0.0555042 ||align=right|     17.0 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 3}{2 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|      18 || 0.0000925 ||align=right| 10,809.0 : 1
 * align=right|      18 || 0.0000925 ||align=right| 10,809.0 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      60 || 0.0003084 ||align=right|  3,242.0 : 1
 * align=right|      72 || 0.0003700 ||align=right|  2,701.5 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 3}{51-4r \choose 1} \end{matrix}$$
 * align=right|      57 || 0.0002929 ||align=right|  3,412.7 : 1
 * align=right|      45 || 0.0002313 ||align=right|  4,323.0 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 2}^2 \end{matrix}$$
 * align=right|      27 || 0.0001388 ||align=right|  7,205.7 : 1
 * align=right|      27 || 0.0001388 ||align=right|  7,205.7 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|   1,026 || 0.0052729 ||align=right|    188.6 : 1
 * align=right|     810 || 0.0041628 ||align=right|    239.2 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     270 || 0.0013876 ||align=right|    719.7 : 1
 * align=right|     324 || 0.0016651 ||align=right|    599.6 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|   3,420 || 0.0175763 ||align=right|     55.9 : 1
 * align=right|   3,240 || 0.0166512 ||align=right|     59.1 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     180 || 0.0009251 ||align=right|  1,080.0 : 1
 * align=right|     216 || 0.0011101 ||align=right|    899.8 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|   5,130 || 0.0263645 ||align=right|     36.9 : 1
 * align=right|   4,860 || 0.0249769 ||align=right|     39.0 : 1
 * $$\begin{matrix} {4r-3 \choose 2}{51-4r \choose 2} \end{matrix}$$
 * align=right|  69,426 || 0.3567993 ||align=right|      1.8 : 1
 * align=right|  55,440 || 0.2849214 ||align=right|      2.5 : 1
 * $$\begin{matrix} {4r-3 \choose 1}{51-4r \choose 3} \end{matrix}$$
 * align=right|  28,101 || 0.1444187 ||align=right|      5.9 : 1
 * align=right|  15,015 || 0.0771662 ||align=right|     12.0 : 1
 * $$\begin{matrix} {51-4r \choose 4} \end{matrix}$$
 * align=right|   3,876 || 0.0199198 ||align=right|     49.2 : 1
 * align=right|   1,365 || 0.0070151 ||align=right|    141.5 : 1
 * colspan=2| Total
 * align=right| 121,776 || 0.6258403 ||align=right|      0.6 : 1
 * align=right|  93,618 || 0.4811286 ||align=right|      1.1 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=24| Make low ! ABCDE ! AABCD ! ABCD* ! AABBC ! AABC* ! AAABC ! ABC** ! xABCD ! xABC* ! xAABC ! xxABC ! xyABC ! xxxAB ! xxyAB ! xxAAB ! xxAB* ! xyAAB ! xyAB* ! xAAAB ! xAABB ! xAAB* ! xAB** ! xyzAB ! rowspan=15| Miss low ! xxxyA ! xxyyA ! xxyAA ! xxyA* ! xyAAA ! xyAA* ! xyA** ! xxxyz ! xxyyz ! xxyzA ! xxyz* ! xyzAA ! xyzA* ! xyz** ! rowspan=24| No low ! AAAAB ! AAAA* ! AAABB ! AAAB* ! AABB* ! AAA** ! AAB** ! xxxyy ! xxxy* ! xxxAA ! xxxA* ! xxx** ! xxyy* ! xxy** ! xxAAA ! xxAA* ! xxA** ! xAAAA ! xAAA* ! xAA** ! ??*** ! ?**** ! *****
 * +Derivations for low hand shape xyz* (three low ranks and a high card) on the river
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-3 \choose 5}{4 \choose 1}^5 \end{matrix}$$
 * align=right|   1,024 || 0.0005980 ||align=right|  1,671.2 : 1
 * align=right|   6,144 || 0.0035881 ||align=right|    277.7 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   7,680 || 0.0044852 ||align=right|    222.0 : 1
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * $$\begin{matrix} {r-3 \choose 4}{4 \choose 1}^4{51-4r \choose 1} \end{matrix}$$
 * align=right|  24,320 || 0.0142031 ||align=right|     69.4 : 1
 * align=right|  57,600 || 0.0336389 ||align=right|     28.7 : 1
 * $$\begin{matrix} {r-3 \choose 2}{4 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * align=right|   8,640 || 0.0050458 ||align=right|    197.2 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|  54,720 || 0.0319569 ||align=right|     30.3 : 1
 * align=right|  86,400 || 0.0504583 ||align=right|     18.8 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,920 || 0.0011213 ||align=right|    890.8 : 1
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * $$\begin{matrix} {r-3 \choose 3}{4 \choose 1}^3{51-4r \choose 2} \end{matrix}$$
 * align=right| 109,440 || 0.0639139 ||align=right|     14.6 : 1
 * align=right| 134,400 || 0.0784907 ||align=right|     11.7 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|  11,520 || 0.0067278 ||align=right|    147.6 : 1
 * align=right|  34,560 || 0.0201833 ||align=right|     48.5 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^3{51-4r \choose 1} \end{matrix}$$
 * align=right| 109,440 || 0.0639139 ||align=right|     14.6 : 1
 * align=right| 172,800 || 0.1009167 ||align=right|      8.9 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  25,920 || 0.0151375 ||align=right|     65.1 : 1
 * align=right|  51,840 || 0.0302750 ||align=right|     32.0 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   5,760 || 0.0033639 ||align=right|    296.3 : 1
 * align=right|  11,520 || 0.0067278 ||align=right|    147.6 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|  17,280 || 0.0100917 ||align=right|     98.1 : 1
 * align=right|  34,560 || 0.0201833 ||align=right|     48.5 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 3}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     480 || 0.0002803 ||align=right|  3,566.3 : 1
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   8,640 || 0.0050458 ||align=right|    197.2 : 1
 * align=right|  12,960 || 0.0075687 ||align=right|    131.1 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * align=right|   6,480 || 0.0037844 ||align=right|    263.2 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|  27,360 || 0.0159785 ||align=right|     61.6 : 1
 * align=right|  32,400 || 0.0189219 ||align=right|     51.8 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|  12,960 || 0.0075687 ||align=right|    131.1 : 1
 * align=right|  19,440 || 0.0113531 ||align=right|     87.1 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 2}{4 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|  82,080 || 0.0479354 ||align=right|     19.9 : 1
 * align=right|  97,200 || 0.0567656 ||align=right|     16.6 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,880 || 0.0016819 ||align=right|    593.6 : 1
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|   3,240 || 0.0018922 ||align=right|    527.5 : 1
 * align=right|   4,860 || 0.0028383 ||align=right|    351.3 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|  82,080 || 0.0479354 ||align=right|     19.9 : 1
 * align=right|  97,200 || 0.0567656 ||align=right|     16.6 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{51-4r \choose 2} \end{matrix}$$
 * align=right| 246,240 || 0.1438062 ||align=right|      6.0 : 1
 * align=right| 226,800 || 0.1324531 ||align=right|      6.5 : 1
 * $$\begin{matrix} {3 \choose 3}{3 \choose 1}^3{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * align=right|   6,480 || 0.0037844 ||align=right|    263.2 : 1
 * colspan=2| Total
 * align=right| 847,944 || 0.4952065 ||align=right|      1.0 : 1
 * align=right|1,134,204 || 0.6623847 ||align=right|     0.5 : 1
 * align=right|1,134,204 || 0.6623847 ||align=right|     0.5 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 3}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     360 || 0.0002102 ||align=right|  4,755.4 : 1
 * align=right|     432 || 0.0002523 ||align=right|  3,962.7 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 2}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     540 || 0.0003154 ||align=right|  3,169.9 : 1
 * align=right|     648 || 0.0003784 ||align=right|  2,641.4 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|   1,620 || 0.0009461 ||align=right|  1,056.0 : 1
 * align=right|   1,944 || 0.0011353 ||align=right|    879.8 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|  20,520 || 0.0119839 ||align=right|     82.4 : 1
 * align=right|  19,440 || 0.0113531 ||align=right|     87.1 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     540 || 0.0003154 ||align=right|  3,169.9 : 1
 * align=right|     648 || 0.0003784 ||align=right|  2,641.4 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|  15,390 || 0.0089879 ||align=right|    110.3 : 1
 * align=right|  14,580 || 0.0085148 ||align=right|    116.4 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$$
 * align=right|  92,340 || 0.0539273 ||align=right|     17.5 : 1
 * align=right|  68,040 || 0.0397359 ||align=right|     24.2 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 3}{2 \choose 2}{3 \choose 1}^2 \end{matrix}$$
 * align=right|      27 || 0.0000158 ||align=right| 63,417.7 : 1
 * align=right|      27 || 0.0000158 ||align=right| 63,417.7 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 2}^2{1 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|      81 || 0.0000473 ||align=right| 21,138.6 : 1
 * align=right|      81 || 0.0000473 ||align=right| 21,138.6 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 2}{3 \choose 1}^2 \end{matrix}$$
 * align=right|   1,620 || 0.0009461 ||align=right|  1,056.0 : 1
 * align=right|   1,944 || 0.0011353 ||align=right|    879.8 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 2}{3 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|   1,539 || 0.0008988 ||align=right|  1,111.6 : 1
 * align=right|   1,215 || 0.0007096 ||align=right|  1,408.3 : 1
 * $$\begin{matrix} {3 \choose 3}{3 \choose 1}^3{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     810 || 0.0004730 ||align=right|  2,113.0 : 1
 * align=right|     972 || 0.0005677 ||align=right|  1,760.6 : 1
 * $$\begin{matrix} {3 \choose 3}{3 \choose 1}^3{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|  10,260 || 0.0059919 ||align=right|    165.9 : 1
 * align=right|   9,720 || 0.0056766 ||align=right|    175.2 : 1
 * $$\begin{matrix} {3 \choose 3}{3 \choose 1}^3{51-4r \choose 2} \end{matrix}$$
 * align=right|   4,617 || 0.0026964 ||align=right|    369.9 : 1
 * align=right|   2,835 || 0.0016557 ||align=right|    603.0 : 1
 * colspan=2| Total
 * align=right| 150,264 || 0.0877554 ||align=right|     10.4 : 1
 * align=right| 122,526 || 0.0715562 ||align=right|     13.0 : 1
 * align=right| 122,526 || 0.0715562 ||align=right|     13.0 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 4}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      80 || 0.0000467 ||align=right| 21,402.8 : 1
 * align=right|     120 || 0.0000701 ||align=right| 14,268.2 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 4}{51-4r \choose 1} \end{matrix}$$
 * align=right|      95 || 0.0000555 ||align=right| 18,023.3 : 1
 * align=right|      90 || 0.0000526 ||align=right| 19,024.6 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     480 || 0.0002803 ||align=right|  3,566.3 : 1
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|   6,080 || 0.0035508 ||align=right|    280.6 : 1
 * align=right|   7,200 || 0.0042049 ||align=right|    236.8 : 1
 * $$\begin{matrix} {r-3 \choose 2}{4 \choose 2}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|   6,840 || 0.0039946 ||align=right|    249.3 : 1
 * align=right|   8,100 || 0.0047305 ||align=right|    210.4 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3}{51-4r \choose 2} \end{matrix}$$
 * align=right|   3,420 || 0.0019973 ||align=right|    499.7 : 1
 * align=right|   2,520 || 0.0014717 ||align=right|    678.5 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$$
 * align=right|  82,080 || 0.0479354 ||align=right|     19.9 : 1
 * align=right|  75,600 || 0.0441510 ||align=right|     21.6 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 3}{2 \choose 1}{3 \choose 2} \end{matrix}$$
 * align=right|      18 || 0.0000105 ||align=right| 95,127.0 : 1
 * align=right|      18 || 0.0000105 ||align=right| 95,127.0 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 3}{2 \choose 1}{3 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|     342 || 0.0001997 ||align=right|  5,005.7 : 1
 * align=right|     270 || 0.0001577 ||align=right|  6,340.9 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      90 || 0.0000526 ||align=right| 19,024.6 : 1
 * align=right|     108 || 0.0000631 ||align=right| 15,853.7 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 1} \end{matrix}$$
 * align=right|   1,140 || 0.0006658 ||align=right|  1,501.0 : 1
 * align=right|   1,080 || 0.0006307 ||align=right|  1,584.5 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 3}{51-4r \choose 2} \end{matrix}$$
 * align=right|     513 || 0.0002996 ||align=right|  3,336.8 : 1
 * align=right|     315 || 0.0001840 ||align=right|  5,434.9 : 1
 * $$\begin{matrix} {3 \choose 2}{3 \choose 2}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|     513 || 0.0002996 ||align=right|  3,336.8 : 1
 * align=right|     405 || 0.0002365 ||align=right|  4,226.9 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{2 \choose 1}{3 \choose 1}{51-4r \choose 2} \end{matrix}$$
 * align=right|   9,234 || 0.0053927 ||align=right|    184.4 : 1
 * align=right|   5,670 || 0.0033113 ||align=right|    301.0 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     180 || 0.0001051 ||align=right|  9,511.8 : 1
 * align=right|     216 || 0.0001261 ||align=right|  7,926.3 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{51-4r \choose 1} \end{matrix}$$
 * align=right|   5,130 || 0.0029960 ||align=right|    332.8 : 1
 * align=right|   4,860 || 0.0028383 ||align=right|    351.3 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}{51-4r \choose 2} \end{matrix}$$
 * align=right|  30,780 || 0.0179758 ||align=right|     54.6 : 1
 * align=right|  22,680 || 0.0132453 ||align=right|     74.5 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|      45 || 0.0000263 ||align=right| 38,050.2 : 1
 * align=right|      54 || 0.0000315 ||align=right| 31,708.3 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3}{51-4r \choose 1} \end{matrix}$$
 * align=right|   3,420 || 0.0019973 ||align=right|    499.7 : 1
 * align=right|   3,240 || 0.0018922 ||align=right|    527.5 : 1
 * $$\begin{matrix} {3 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{51-4r \choose 2} \end{matrix}$$
 * align=right|  46,170 || 0.0269637 ||align=right|     36.1 : 1
 * align=right|  34,020 || 0.0198680 ||align=right|     49.3 : 1
 * $$\begin{matrix} {4r-3 \choose 2}{51-4r \choose 3} \end{matrix}$$
 * align=right| 393,414 || 0.2297571 ||align=right|      3.4 : 1
 * align=right| 240,240 || 0.1403022 ||align=right|      6.1 : 1
 * $$\begin{matrix} {4r-3 \choose 1}{51-4r \choose 4} \end{matrix}$$
 * align=right| 112,404 || 0.0656449 ||align=right|     14.2 : 1
 * align=right|  45,045 || 0.0263067 ||align=right|     37.0 : 1
 * $$\begin{matrix} {51-4r \choose 5} \end{matrix}$$
 * align=right|  11,628 || 0.0067909 ||align=right|    146.3 : 1
 * align=right|   3,003 || 0.0017538 ||align=right|    569.2 : 1
 * colspan=2| Total
 * align=right| 714,096 || 0.4170381 ||align=right|      1.4 : 1
 * align=right| 455,574 || 0.2660591 ||align=right|      2.8 : 1
 * }
 * }

Derivations for making low hands with a low pair and two low ranks
Starting hands with a low pair and two low ranks (xxyz) can make a low hand when either at least three other low ranks beside x, y and z appear on the board; or when two other low ranks beside x, y and z and at least one of the low ranks x, y or z appears. With no high cards in the hand, there are $$52 - 4r - 0 = 52 - 4r$$ high cards and $$4r - 4 + 0 = 4r - 4$$ low cards left in the deck. The following tables show the derivations when holding a low pair and two low ranks.


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=4| Make low ! ABC ! xAB ! yAB ! rowspan=4| Miss low ! xyA ! yzA ! xyz ! rowspan=14| No low ! AAA ! AAB ! xxy ! xxA ! xyy ! xAA ! yyy ! yyA ! yAA ! yyz ! ??* ! ?** ! ***
 * +Derivations for low hand shape xxyz (a low pair and two low ranks) on the flop
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|     640 || 0.0370028 ||align=right|     26.0 : 1
 * align=right|   1,280 || 0.0740056 ||align=right|     12.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     320 || 0.0185014 ||align=right|     53.1 : 1
 * align=right|     480 || 0.0277521 ||align=right|     35.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     960 || 0.0555042 ||align=right|     17.0 : 1
 * align=right|   1,440 || 0.0832562 ||align=right|     11.0 : 1
 * colspan=2| Total
 * align=right|   1,920 || 0.1110083 ||align=right|      8.0 : 1
 * align=right|   3,200 || 0.1850139 ||align=right|      4.4 : 1
 * align=right|   3,200 || 0.1850139 ||align=right|      4.4 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     240 || 0.0138760 ||align=right|     71.1 : 1
 * align=right|     288 || 0.0166512 ||align=right|     95.1 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     180 || 0.0104070 ||align=right|     59.1 : 1
 * align=right|     216 || 0.0124884 ||align=right|     79.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^2 \end{matrix}$$
 * align=right|      18 || 0.0010407 ||align=right|    959.9 : 1
 * align=right|      18 || 0.0010407 ||align=right|    959.9 : 1
 * colspan=2| Total
 * align=right|     438 || 0.0253238 ||align=right|     38.5 : 1
 * align=right|     522 || 0.0301804 ||align=right|     32.1 : 1
 * align=right|     522 || 0.0301804 ||align=right|     32.1 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      20 || 0.0011563 ||align=right|    863.8 : 1
 * align=right|      24 || 0.0013876 ||align=right|    719.7 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     480 || 0.0277521 ||align=right|     35.0 : 1
 * align=right|     720 || 0.0416281 ||align=right|     23.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|       6 || 0.0003469 ||align=right|  2,881.7 : 1
 * align=right|       6 || 0.0003469 ||align=right|  2,881.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      20 || 0.0011563 ||align=right|    863.8 : 1
 * align=right|      24 || 0.0013876 ||align=right|    719.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2} \end{matrix}$$
 * align=right|      12 || 0.0006938 ||align=right|  1,440.3 : 1
 * align=right|      12 || 0.0006938 ||align=right|  1,440.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      60 || 0.0034690 ||align=right|    287.3 : 1
 * align=right|      72 || 0.0041628 ||align=right|    239.2 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3} \end{matrix}$$
 * align=right|       2 || 0.0001156 ||align=right|  8,647.0 : 1
 * align=right|       2 || 0.0001156 ||align=right|  8,647.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     120 || 0.0069380 ||align=right|    143.1 : 1
 * align=right|     144 || 0.0083256 ||align=right|    119.1 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     180 || 0.0104070 ||align=right|     59.1 : 1
 * align=right|     216 || 0.0124884 ||align=right|     79.1 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|      18 || 0.0010407 ||align=right|    959.9 : 1
 * align=right|      18 || 0.0010407 ||align=right|    959.9 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   7,560 || 0.4370953 ||align=right|      1.3 : 1
 * align=right|   7,936 || 0.4588344 ||align=right|      1.2 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|   5,320 || 0.3075856 ||align=right|      2.3 : 1
 * align=right|   3,840 || 0.2220167 ||align=right|      3.5 : 1
 * $$\begin{matrix} {52-4r \choose 3} \end{matrix}$$
 * align=right|   1,140 || 0.0659112 ||align=right|     14.2 : 1
 * align=right|     560 || 0.0323774 ||align=right|     29.9 : 1
 * colspan=2| Total
 * align=right|  14,938 || 0.8636679 ||align=right|      0.2 : 1
 * align=right|  13,574 || 0.7848057 ||align=right|      0.3 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=14| Make low ! ABCD ! AABC ! ABC* ! xABC ! yABC ! xxAB ! xAAB ! xyAB ! xAB* ! yyAB ! yAAB ! yAB* ! yzAB ! rowspan=12| Miss low ! xxyA ! xyyA ! xyAA ! xyA* ! yyzA ! yzAA ! yzA* ! xxyz ! xyyz ! xyzA ! xyz* ! rowspan=26| No low ! AAAA ! AAAB ! AABB ! AAA* ! AAB* ! xxyy ! xxy* ! xxAA ! xxA* ! xyyy ! xyy* ! xAAA ! xAA* ! yyyA ! yyy* ! yyAA ! yyA* ! yAAA ! yAA* ! yyyz ! yyzz ! yyz* ! ??** ! ?*** ! ****
 * +Derivations for low hand shape xxyz (a low pair and two low ranks) on the turn
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-3 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   1,280 || 0.0065783 ||align=right|    151.0 : 1
 * align=right|   3,840 || 0.0197348 ||align=right|     49.7 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   2,880 || 0.0148011 ||align=right|     66.6 : 1
 * align=right|   5,760 || 0.0296022 ||align=right|     32.8 : 1
 * $$\begin{matrix} {r-3 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$$
 * align=right|  12,800 || 0.0657827 ||align=right|     14.2 : 1
 * align=right|  20,480 || 0.1052523 ||align=right|      8.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   1,280 || 0.0065783 ||align=right|    151.0 : 1
 * align=right|   2,560 || 0.0131565 ||align=right|     75.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   6,720 || 0.0345359 ||align=right|     28.0 : 1
 * align=right|   7,680 || 0.0394696 ||align=right|     24.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     160 || 0.0008223 ||align=right|  1,215.1 : 1
 * align=right|     240 || 0.0012334 ||align=right|    809.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     960 || 0.0049337 ||align=right|    201.7 : 1
 * align=right|   1,440 || 0.0074006 ||align=right|    134.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,920 || 0.0098674 ||align=right|    100.3 : 1
 * align=right|   2,880 || 0.0148011 ||align=right|     66.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|   6,400 || 0.0328914 ||align=right|     29.4 : 1
 * align=right|   7,680 || 0.0394696 ||align=right|     24.3 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     960 || 0.0049337 ||align=right|    201.7 : 1
 * align=right|   1,440 || 0.0074006 ||align=right|    134.1 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,880 || 0.0148011 ||align=right|     66.6 : 1
 * align=right|   4,320 || 0.0222017 ||align=right|     44.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  19,200 || 0.0986741 ||align=right|      9.1 : 1
 * align=right|  23,040 || 0.1184089 ||align=right|      7.4 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,440 || 0.0074006 ||align=right|    134.1 : 1
 * align=right|   2,160 || 0.0111008 ||align=right|     89.1 : 1
 * colspan=2| Total
 * align=right|  56,000 || 0.2877994 ||align=right|      2.5 : 1
 * align=right|  83,520 || 0.4292322 ||align=right|      1.3 : 1
 * align=right|  83,520 || 0.4292322 ||align=right|      1.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     120 || 0.0006167 ||align=right|  1,620.5 : 1
 * align=right|     144 || 0.0007401 ||align=right|  1,350.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     240 || 0.0012334 ||align=right|    809.8 : 1
 * align=right|     288 || 0.0014801 ||align=right|    674.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     360 || 0.0018501 ||align=right|    539.5 : 1
 * align=right|     432 || 0.0022202 ||align=right|    449.4 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   4,800 || 0.0246685 ||align=right|     39.5 : 1
 * align=right|   4,608 || 0.0236818 ||align=right|     41.2 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     360 || 0.0018501 ||align=right|    539.5 : 1
 * align=right|     432 || 0.0022202 ||align=right|    449.4 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     270 || 0.0013876 ||align=right|    719.7 : 1
 * align=right|     324 || 0.0016651 ||align=right|    599.6 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   3,600 || 0.0185014 ||align=right|     53.1 : 1
 * align=right|   3,456 || 0.0177613 ||align=right|     55.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 2}{3 \choose 1}^2 \end{matrix}$$
 * align=right|       9 || 0.0000463 ||align=right| 21,619.0 : 1
 * align=right|       9 || 0.0000463 ||align=right| 21,619.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|      36 || 0.0001850 ||align=right|  5,404.0 : 1
 * align=right|      36 || 0.0001850 ||align=right|  5,404.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     360 || 0.0018501 ||align=right|    539.5 : 1
 * align=right|     432 || 0.0022202 ||align=right|    449.4 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^2{51-4r \choose 1} \end{matrix}$$
 * align=right|     360 || 0.0018501 ||align=right|    539.5 : 1
 * align=right|     288 || 0.0014801 ||align=right|    674.6 : 1
 * colspan=2| Total
 * align=right|  10,515 || 0.0540395 ||align=right|     17.5 : 1
 * align=right|  10,449 || 0.0537003 ||align=right|     17.6 : 1
 * align=right|  10,449 || 0.0537003 ||align=right|     17.6 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       5 || 0.0000257 ||align=right| 38,915.0 : 1
 * align=right|       6 || 0.0000308 ||align=right| 32,429.0 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     320 || 0.0016446 ||align=right|    607.1 : 1
 * align=right|     480 || 0.0024669 ||align=right|    404.4 : 1
 * $$\begin{matrix} {r-3 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     360 || 0.0018501 ||align=right|    539.5 : 1
 * align=right|     540 || 0.0027752 ||align=right|    359.3 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|     400 || 0.0020557 ||align=right|    485.5 : 1
 * align=right|     384 || 0.0019735 ||align=right|    505.7 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   9,600 || 0.0493370 ||align=right|     19.3 : 1
 * align=right|  11,520 || 0.0592044 ||align=right|     15.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 2} \end{matrix}$$
 * align=right|       6 || 0.0000308 ||align=right| 32,429.0 : 1
 * align=right|       6 || 0.0000308 ||align=right| 32,429.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|     120 || 0.0006167 ||align=right|  1,620.5 : 1
 * align=right|      96 || 0.0004934 ||align=right|  2,025.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      30 || 0.0001542 ||align=right|  6,485.0 : 1
 * align=right|      36 || 0.0001850 ||align=right|  5,404.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|     400 || 0.0020557 ||align=right|    485.5 : 1
 * align=right|     384 || 0.0019735 ||align=right|    505.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 3} \end{matrix}$$
 * align=right|       4 || 0.0000206 ||align=right| 48,644.0 : 1
 * align=right|       4 || 0.0000206 ||align=right| 48,644.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|     240 || 0.0012334 ||align=right|    809.8 : 1
 * align=right|     192 || 0.0009867 ||align=right|  1,012.4 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      40 || 0.0002056 ||align=right|  4,863.5 : 1
 * align=right|      48 || 0.0002467 ||align=right|  4,052.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   1,200 || 0.0061671 ||align=right|    161.2 : 1
 * align=right|   1,152 || 0.0059204 ||align=right|    167.9 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      40 || 0.0002056 ||align=right|  4,863.5 : 1
 * align=right|      48 || 0.0002467 ||align=right|  4,052.8 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|      40 || 0.0002056 ||align=right|  4,863.5 : 1
 * align=right|      32 || 0.0001645 ||align=right|  6,079.6 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     180 || 0.0009251 ||align=right|  1,080.0 : 1
 * align=right|     216 || 0.0011101 ||align=right|    899.8 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   2,400 || 0.0123343 ||align=right|     80.1 : 1
 * align=right|   2,304 || 0.0118409 ||align=right|     83.5 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     120 || 0.0006167 ||align=right|  1,620.5 : 1
 * align=right|     144 || 0.0007401 ||align=right|  1,350.3 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   3,600 || 0.0185014 ||align=right|     53.1 : 1
 * align=right|   3,456 || 0.0177613 ||align=right|     55.3 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|       6 || 0.0000308 ||align=right| 32,429.0 : 1
 * align=right|       6 || 0.0000308 ||align=right| 32,429.0 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 2}^2 \end{matrix}$$
 * align=right|       9 || 0.0000463 ||align=right| 21,619.0 : 1
 * align=right|       9 || 0.0000463 ||align=right| 21,619.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|     360 || 0.0018501 ||align=right|    539.5 : 1
 * align=right|     288 || 0.0014801 ||align=right|    674.6 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 2} \end{matrix}$$
 * align=right|  71,820 || 0.3691027 ||align=right|      1.7 : 1
 * align=right|  59,520 || 0.3058896 ||align=right|      2.3 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 3} \end{matrix}$$
 * align=right|  31,920 || 0.1640456 ||align=right|      5.1 : 1
 * align=right|  17,920 || 0.0920958 ||align=right|      9.9 : 1
 * $$\begin{matrix} {52-4r \choose 4} \end{matrix}$$
 * align=right|   4,845 || 0.0248998 ||align=right|     39.2 : 1
 * align=right|   1,820 || 0.0093535 ||align=right|    105.9 : 1
 * colspan=2| Total
 * align=right| 128,065 || 0.6581612 ||align=right|      0.5 : 1
 * align=right| 100,611 || 0.5170675 ||align=right|      0.9 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=39| Make low ! ABCDE ! AABCD ! ABCD* ! AABBC ! AABC* ! AAABC ! ABC** ! xABCD ! xABC* ! xAABC ! xxABC ! xyABC ! yyABC ! yABCD ! yABC* ! yAABC ! xxyAB ! xxAAB ! xxAB* ! xyyAB ! xyAAB ! xyAB* ! xAAAB ! xAABB ! xAAB* ! xAB** ! yyyAB ! yyAAB ! yyAB* ! yAAAB ! yAABB ! yAAB* ! yAB** ! yyzAB ! yzAAB ! yzABC ! yzAB* ! xyzAB ! rowspan=27| Miss low ! xxyyA ! xxyAA ! xxyA* ! xyyyA ! xyyAA ! xyyA* ! xyAAA ! xyAA* ! xyA** ! yyyzA ! yyzzA ! yyzAA ! yyzA* ! yzAAA ! yzAA* ! yzA** ! xxyyz ! xxyzA ! xxyz* ! xyyyz ! xyyzz ! xyyzA ! xyyz* ! xyzAA ! xyzA* ! xyz** ! rowspan=35| No low ! AAAAB ! AAAA* ! AAABB ! AAAB* ! AABB* ! AAA** ! AAB** ! xxyyy ! xxyy* ! xxy** ! xxAAA ! xxAA* ! xxA** ! xyyy* ! xyy** ! xAAAA ! xAAA* ! xAA** ! yyyAA ! yyyA* ! yyy** ! yyAAA ! yyAA* ! yyA** ! yAAAA ! yAAA* ! yAA** ! yyyzz ! yyyz* ! yyzz* ! yyz** ! ??*** ! ?**** ! *****
 * +Derivations for low hand shape xxyz (a low pair and two low ranks) on the river
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-3 \choose 5}{4 \choose 1}^5 \end{matrix}$$
 * align=right|   1,024 || 0.0005980 ||align=right|  1,671.2 : 1
 * align=right|   6,144 || 0.0035881 ||align=right|    277.7 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   7,680 || 0.0044852 ||align=right|     30.9 : 1
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * $$\begin{matrix} {r-3 \choose 4}{4 \choose 1}^4{52-4r \choose 1} \end{matrix}$$
 * align=right|  25,600 || 0.0149506 ||align=right|    222.0 : 1
 * align=right|  61,440 || 0.0358815 ||align=right|     26.9 : 1
 * $$\begin{matrix} {r-3 \choose 2}{4 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * align=right|   8,640 || 0.0050458 ||align=right|    197.2 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  57,600 || 0.0336389 ||align=right|     28.7 : 1
 * align=right|  92,160 || 0.0538222 ||align=right|     17.6 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,920 || 0.0011213 ||align=right|    890.8 : 1
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * $$\begin{matrix} {r-3 \choose 3}{4 \choose 1}^3{52-4r \choose 2} \end{matrix}$$
 * align=right| 121,600 || 0.0710154 ||align=right|     13.1 : 1
 * align=right| 153,600 || 0.0897037 ||align=right|     10.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   2,560 || 0.0014951 ||align=right|    667.9 : 1
 * align=right|   7,680 || 0.0044852 ||align=right|    222.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$$
 * align=right|  25,600 || 0.0149506 ||align=right|     65.9 : 1
 * align=right|  40,960 || 0.0239210 ||align=right|     40.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   5,760 || 0.0033639 ||align=right|    296.3 : 1
 * align=right|  11,520 || 0.0067278 ||align=right|    147.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|     640 || 0.0003738 ||align=right|  2,674.5 : 1
 * align=right|   1,280 || 0.0007475 ||align=right|  1,336.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   7,680 || 0.0044852 ||align=right|    222.0 : 1
 * align=right|  15,360 || 0.0089704 ||align=right|    110.5 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * align=right|   7,680 || 0.0044852 ||align=right|    222.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   7,680 || 0.0044852 ||align=right|    222.0 : 1
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$$
 * align=right|  76,800 || 0.0448518 ||align=right|     21.3 : 1
 * align=right| 122,880 || 0.0717630 ||align=right|     12.9 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  17,280 || 0.0100917 ||align=right|     98.1 : 1
 * align=right|  34,560 || 0.0201833 ||align=right|     48.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     960 || 0.0005606 ||align=right|  1,782.7 : 1
 * align=right|   1,440 || 0.0008410 ||align=right|  1,188.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     480 || 0.0002803 ||align=right|  3,566.3 : 1
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|   3,200 || 0.0018688 ||align=right|    534.1 : 1
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,920 || 0.0011213 ||align=right|    890.8 : 1
 * align=right|   2,880 || 0.0016819 ||align=right|    593.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   5,760 || 0.0033639 ||align=right|    296.3 : 1
 * align=right|   8,640 || 0.0050458 ||align=right|    197.2 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  38,400 || 0.0224259 ||align=right|     43.6 : 1
 * align=right|  46,080 || 0.0269111 ||align=right|     36.2 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     640 || 0.0003738 ||align=right|  2,674.5 : 1
 * align=right|     960 || 0.0005606 ||align=right|  1,782.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * align=right|   1,080 || 0.0006307 ||align=right|  1,584.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|  19,200 || 0.0112130 ||align=right|     88.2 : 1
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{52-4r \choose 2} \end{matrix}$$
 * align=right|  60,800 || 0.0355077 ||align=right|     27.2 : 1
 * align=right|  57,600 || 0.0336389 ||align=right|     28.7 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     320 || 0.0001869 ||align=right|  5,350.0 : 1
 * align=right|     480 || 0.0002803 ||align=right|  3,566.3 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,880 || 0.0016819 ||align=right|    593.6 : 1
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  19,200 || 0.0112130 ||align=right|     88.2 : 1
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,920 || 0.0011213 ||align=right|    890.8 : 1
 * align=right|   2,880 || 0.0016819 ||align=right|    593.6 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|   1,620 || 0.0009461 ||align=right|  1,056.0 : 1
 * align=right|   3,240 || 0.0018922 ||align=right|    527.5 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|  57,600 || 0.0336389 ||align=right|     28.7 : 1
 * align=right|  69,120 || 0.0403667 ||align=right|     23.8 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{52-4r \choose 2} \end{matrix}$$
 * align=right| 182,400 || 0.1065231 ||align=right|      8.4 : 1
 * align=right| 172,800 || 0.1009167 ||align=right|      8.9 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   2,880 || 0.0016819 ||align=right|    593.6 : 1
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * align=right|   6,480 || 0.0037844 ||align=right|    263.2 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   5,760 || 0.0033639 ||align=right|    296.3 : 1
 * align=right|  11,520 || 0.0067278 ||align=right|    147.6 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  28,800 || 0.0168194 ||align=right|     58.5 : 1
 * align=right|  34,560 || 0.0201833 ||align=right|     48.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 2}{3 \choose 1}^2{r-3 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   2,880 || 0.0016819 ||align=right|    593.6 : 1
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * colspan=2| Total
 * align=right| 810,784 || 0.4735047 ||align=right|      1.1 : 1
 * align=right|1,097,184 || 0.6407647 ||align=right|     0.6 : 1
 * align=right|1,097,184 || 0.6407647 ||align=right|     0.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     120 || 0.0000701 ||align=right| 14,268.2 : 1
 * align=right|     144 || 0.0000841 ||align=right| 11,890.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     180 || 0.0001051 ||align=right|  9,511.8 : 1
 * align=right|     216 || 0.0001261 ||align=right|  7,926.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   2,400 || 0.0014016 ||align=right|    712.5 : 1
 * align=right|   2,304 || 0.0013456 ||align=right|    742.2 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      80 || 0.0000467 ||align=right| 21,402.8 : 1
 * align=right|      96 || 0.0000561 ||align=right| 17,835.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     360 || 0.0002102 ||align=right|  4,755.4 : 1
 * align=right|     432 || 0.0002523 ||align=right|  3,962.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   4,800 || 0.0028032 ||align=right|    355.7 : 1
 * align=right|   4,608 || 0.0026911 ||align=right|    370.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     240 || 0.0001402 ||align=right|  7,133.6 : 1
 * align=right|     288 || 0.0001682 ||align=right|  5,944.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   7,200 || 0.0042049 ||align=right|    236.8 : 1
 * align=right|   6,912 || 0.0040367 ||align=right|    246.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|  45,600 || 0.0266308 ||align=right|     36.6 : 1
 * align=right|  34,560 || 0.0201833 ||align=right|     48.5 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     120 || 0.0000701 ||align=right| 14,268.2 : 1
 * align=right|     144 || 0.0000841 ||align=right| 11,890.0 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 2}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     180 || 0.0001051 ||align=right|  9,511.8 : 1
 * align=right|     216 || 0.0001261 ||align=right|  7,926.3 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     540 || 0.0003154 ||align=right|  3,169.9 : 1
 * align=right|     648 || 0.0003784 ||align=right|  2,641.4 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   7,200 || 0.0042049 ||align=right|    236.8 : 1
 * align=right|   6,912 || 0.0040367 ||align=right|    246.7 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     180 || 0.0001051 ||align=right|  9,511.8 : 1
 * align=right|     216 || 0.0001261 ||align=right|  7,926.3 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   5,400 || 0.0031536 ||align=right|    316.1 : 1
 * align=right|   5,184 || 0.0030275 ||align=right|    329.3 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|  34,200 || 0.0199731 ||align=right|     49.1 : 1
 * align=right|  25,920 || 0.0151375 ||align=right|     65.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|      18 || 0.0000105 ||align=right| 95,127.0 : 1
 * align=right|      18 || 0.0000105 ||align=right| 95,127.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     180 || 0.0001051 ||align=right|  9,511.8 : 1
 * align=right|     216 || 0.0001261 ||align=right|  7,926.3 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 2}{3 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|     180 || 0.0001051 ||align=right|  9,511.8 : 1
 * align=right|     144 || 0.0000841 ||align=right| 11,890.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|      12 || 0.0000070 ||align=right|  14,2691 : 1
 * align=right|      12 || 0.0000070 ||align=right|  14,2691 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 2}^2 \end{matrix}$$
 * align=right|      18 || 0.0000105 ||align=right| 95,127.0 : 1
 * align=right|      18 || 0.0000105 ||align=right| 95,127.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * align=right|     864 || 0.0005046 ||align=right|  1,980.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * align=right|     576 || 0.0003364 ||align=right|  2,971.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     540 || 0.0003154 ||align=right|  3,169.9 : 1
 * align=right|     648 || 0.0003784 ||align=right|  2,641.4 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^2{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   7,200 || 0.0042049 ||align=right|    236.8 : 1
 * align=right|   6,912 || 0.0040367 ||align=right|    246.7 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 2}{3 \choose 1}^2{52-4r \choose 2} \end{matrix}$$
 * align=right|   3,420 || 0.0019973 ||align=right|    499.7 : 1
 * align=right|   2,160 || 0.0012615 ||align=right|    791.7 : 1
 * colspan=2| Total
 * align=right| 121,808 || 0.0711369 ||align=right|     13.1 : 1
 * align=right| 100,368 || 0.0586158 ||align=right|     16.1 : 1
 * align=right| 100,368 || 0.0586158 ||align=right|     16.1 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 4}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      80 || 0.0000467 ||align=right| 21,402.8 : 1
 * align=right|     120 || 0.0000701 ||align=right| 15,019.2 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 4}{52-4r \choose 1} \end{matrix}$$
 * align=right|     100 || 0.0000584 ||align=right| 17,122.0 : 1
 * align=right|      96 || 0.0000561 ||align=right| 17,835.5 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     480 || 0.0002803 ||align=right|  3,566.3 : 1
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   6,400 || 0.0037377 ||align=right|    266.5 : 1
 * align=right|   7,680 || 0.0044852 ||align=right|     30.9 : 1
 * $$\begin{matrix} {r-3 \choose 2}{4 \choose 2}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|   7,200 || 0.0042049 ||align=right|    236.8 : 1
 * align=right|   8,640 || 0.0050458 ||align=right|    197.2 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 3}{52-4r \choose 2} \end{matrix}$$
 * align=right|   3,800 || 0.0022192 ||align=right|    449.6 : 1
 * align=right|   2,880 || 0.0016819 ||align=right|    593.6 : 1
 * $$\begin{matrix} {r-3 \choose 1}{4 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|  91,200 || 0.0532616 ||align=right|     17.8 : 1
 * align=right|  86,400 || 0.0504583 ||align=right|     18.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 3} \end{matrix}$$
 * align=right|       2 || 0.0000012 ||align=right|  856151.0 : 1
 * align=right|       2 || 0.0000012 ||align=right|  856151.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|     120 || 0.0000701 ||align=right| 14,268.2 : 1
 * align=right|      96 || 0.0000561 ||align=right| 17,835.5 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{2 \choose 1}{3 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|   1,140 || 0.0006658 ||align=right|  1,501.0 : 1
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      20 || 0.0000117 ||align=right| 85,614.2 : 1
 * align=right|      24 || 0.0000140 ||align=right| 71,345.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|     600 || 0.0003504 ||align=right|  2,852.8 : 1
 * align=right|     576 || 0.0003364 ||align=right|  2,971.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|   3,800 || 0.0022192 ||align=right|    449.6 : 1
 * align=right|   2,880 || 0.0016819 ||align=right|    593.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|      80 || 0.0000467 ||align=right| 21,402.8 : 1
 * align=right|      64 || 0.0000374 ||align=right| 26,753.8 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{2 \choose 1}{3 \choose 2}{52-4r \choose 2} \end{matrix}$$
 * align=right|   2,280 || 0.0013315 ||align=right|    750.0 : 1
 * align=right|   1,440 || 0.0008410 ||align=right|  1,188.1 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|      10 || 0.0000058 ||align=right|171,229.4 : 1
 * align=right|      12 || 0.0000070 ||align=right|142,691.0 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|     800 || 0.0004672 ||align=right|  2,139.4 : 1
 * align=right|     768 || 0.0004485 ||align=right|  2,228.6 : 1
 * $$\begin{matrix} {1 \choose 1}{2 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 2} \end{matrix}$$
 * align=right|  11,400 || 0.0066577 ||align=right|    149.2 : 1
 * align=right|   8,640 || 0.0050458 ||align=right|    197.2 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      60 || 0.0000350 ||align=right| 28,537.4 : 1
 * align=right|      72 || 0.0000420 ||align=right| 23,781.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|     800 || 0.0004672 ||align=right|  2,139.4 : 1
 * align=right|     768 || 0.0004485 ||align=right|  2,228.6 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{52-4r \choose 2} \end{matrix}$$
 * align=right|     380 || 0.0002219 ||align=right|  4,505.1 : 1
 * align=right|     240 || 0.0001402 ||align=right|  7,133.6 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     120 || 0.0000701 ||align=right| 14,268.2 : 1
 * align=right|     144 || 0.0000841 ||align=right| 11,890.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   3,600 || 0.0021024 ||align=right|    474.6 : 1
 * align=right|   3,456 || 0.0020183 ||align=right|    494.5 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{r-3 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|  22,800 || 0.0133154 ||align=right|     74.1 : 1
 * align=right|  17,280 || 0.0100917 ||align=right|     98.1 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|      30 || 0.0000175 ||align=right| 57,075.8 : 1
 * align=right|      36 || 0.0000210 ||align=right| 47,563.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|   2,400 || 0.0014016 ||align=right|    712.5 : 1
 * align=right|   2,304 || 0.0013456 ||align=right|    742.2 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 1}{r-3 \choose 1}{4 \choose 2}{52-4r \choose 2} \end{matrix}$$
 * align=right|  34,200 || 0.0199731 ||align=right|     49.1 : 1
 * align=right|  25,920 || 0.0151375 ||align=right|     65.1 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 2} \end{matrix}$$
 * align=right|       6 || 0.0000035 ||align=right|285,383.0 : 1
 * align=right|       6 || 0.0000035 ||align=right|285,383.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 3}{1 \choose 1}{3 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|     120 || 0.0000701 ||align=right| 14,268.2 : 1
 * align=right|      96 || 0.0000561 ||align=right| 17,835.5 : 1
 * $$\begin{matrix} {2 \choose 2}{3 \choose 2}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|     180 || 0.0001051 ||align=right|  9,511.8 : 1
 * align=right|     144 || 0.0000841 ||align=right| 11,890.0 : 1
 * $$\begin{matrix} {2 \choose 1}{3 \choose 2}{1 \choose 1}{3 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|   3,420 || 0.0019973 ||align=right|    499.7 : 1
 * align=right|   2,160 || 0.0012615 ||align=right|    791.7 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 3} \end{matrix}$$
 * align=right| 430,920 || 0.2516609 ||align=right|      3.0 : 1
 * align=right| 277,760 || 0.1622142 ||align=right|      5.2 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 4} \end{matrix}$$
 * align=right| 135,660 || 0.0792266 ||align=right|     11.6 : 1
 * align=right|  58,240 || 0.0340127 ||align=right|     28.4 : 1
 * $$\begin{matrix} {52-4r \choose 5} \end{matrix}$$
 * align=right|  15,504 || 0.0090545 ||align=right|    109.4 : 1
 * align=right|   4,368 || 0.0025509 ||align=right|    391.0 : 1
 * colspan=2| Total
 * align=right| 779,712 || 0.4553584 ||align=right|      1.2 : 1
 * align=right| 514,752 || 0.3006195 ||align=right|      2.3 : 1
 * }
 * }

Derivations for making low hands with four low ranks
Starting hands with four low ranks (xyzr) can make a low hand when at least three low ranks appear on the board with at least one low rank beside x, y, z or r. With no high cards in the hand, there are $$52 - 4r - 0 = 52 - 4r$$ high cards and $$4r - 4 + 0 = 4r - 4$$ low cards left in the deck. The following tables show the derivations when holding four low ranks.


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=4| Make low ! ABC ! xAB ! xyA ! rowspan=1| Miss low ! xyz ! rowspan=10| No low ! AAA ! AAB ! xxx ! xxy ! xxA ! xAA ! ??* ! ?** ! ***
 * +Derivations for low hand shape xyzr (four low ranks) on the flop
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-4 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|     256 || 0.0148011 ||align=right|     66.6 : 1
 * align=right|     640 || 0.0370028 ||align=right|     26.0 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,152 || 0.0666050 ||align=right|     14.0 : 1
 * align=right|   1,920 || 0.1110083 ||align=right|      8.0 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     864 || 0.0499537 ||align=right|     19.0 : 1
 * align=right|   1,080 || 0.0624422 ||align=right|     15.0 : 1
 * colspan=2| Total
 * align=right|   2,272 || 0.1313599 ||align=right|      6.6 : 1
 * align=right|   3,640 || 0.2104533 ||align=right|      3.8 : 1
 * align=right|   3,640 || 0.2104533 ||align=right|      3.8 : 1
 * $$\begin{matrix} {4 \choose 3}{3 \choose 1}^3 \end{matrix}$$
 * align=right|     108 || 0.0062442 ||align=right|    159.1 : 1
 * align=right|     108 || 0.0062442 ||align=right|    159.1 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|      16 || 0.0009251 ||align=right|  1,080.0 : 1
 * align=right|      20 || 0.0011563 ||align=right|    863.8 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     288 || 0.0166512 ||align=right|     59.1 : 1
 * align=right|     480 || 0.0277521 ||align=right|     35.0 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 3} \end{matrix}$$
 * align=right|       4 || 0.0002313 ||align=right|  4,323.0 : 1
 * align=right|       4 || 0.0002313 ||align=right|  4,323.0 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|     108 || 0.0062442 ||align=right|    159.1 : 1
 * align=right|     108 || 0.0062442 ||align=right|    159.1 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     192 || 0.0111008 ||align=right|     89.1 : 1
 * align=right|     240 || 0.0138760 ||align=right|     71.1 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     288 || 0.0166512 ||align=right|     59.1 : 1
 * align=right|     360 || 0.0208141 ||align=right|     47.0 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   7,560 || 0.4370953 ||align=right|      1.3 : 1
 * align=right|   7,936 || 0.4588344 ||align=right|      1.2 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|   5,320 || 0.3075856 ||align=right|      2.3 : 1
 * align=right|   3,840 || 0.2220167 ||align=right|      3.5 : 1
 * $$\begin{matrix} {52-4r \choose 3} \end{matrix}$$
 * align=right|   1,140 || 0.0659112 ||align=right|     14.2 : 1
 * align=right|     560 || 0.0323774 ||align=right|     29.9 : 1
 * colspan=2| Total
 * align=right|  14,916 || 0.8623959 ||align=right|      0.2 : 1
 * align=right|  13,548 || 0.7833025 ||align=right|      0.3 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=13| Make low ! ABCD ! AABC ! ABC* ! xABC ! xxAB ! xAAB ! xyAB ! xAB* ! xxyA ! xyAA ! xyA* ! xyzA ! rowspan=4| Miss low ! xxyz ! xyzr ! xyz* ! rowspan=18| No low ! AAAA ! AAAB ! AABB ! AAA* ! AAB* ! xxxy ! xxxA ! xxx* ! xxyy ! xxy* ! xxAA ! xxA* ! xAAA ! xAA* ! ??** ! ?*** ! ****
 * +Derivations for low hand shape xyzr (four low ranks) on the turn
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-4 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|     256 || 0.0013157 ||align=right|    759.1 : 1
 * align=right|   1,280 || 0.0065783 ||align=right|    151.0 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 2}{r-5 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,152 || 0.0059204 ||align=right|    167.9 : 1
 * align=right|   2,880 || 0.0148011 ||align=right|     66.6 : 1
 * $$\begin{matrix} {r-4 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$$
 * align=right|   5,120 || 0.0263131 ||align=right|     37.0 : 1
 * align=right|  10,240 || 0.0526262 ||align=right|     18.0 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   3,072 || 0.0157879 ||align=right|     62.3 : 1
 * align=right|   7,680 || 0.0394696 ||align=right|     24.3 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   1,152 || 0.0059204 ||align=right|    167.9 : 1
 * align=right|   1,920 || 0.0098674 ||align=right|    100.3 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   3,456 || 0.0177613 ||align=right|     55.3 : 1
 * align=right|   5,760 || 0.0296022 ||align=right|     32.8 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|   5,184 || 0.0266420 ||align=right|     36.5 : 1
 * align=right|   8,640 || 0.0444033 ||align=right|     21.5 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  23,040 || 0.1184089 ||align=right|      7.4 : 1
 * align=right|  30,720 || 0.1578785 ||align=right|      5.3 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,728 || 0.0088807 ||align=right|    111.6 : 1
 * align=right|   2,160 || 0.0111008 ||align=right|     89.1 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|   1,296 || 0.0066605 ||align=right|    149.1 : 1
 * align=right|   1,620 || 0.0083256 ||align=right|    119.1 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|  17,280 || 0.0888067 ||align=right|     10.3 : 1
 * align=right|  17,280 || 0.0888067 ||align=right|     10.3 : 1
 * $$\begin{matrix} {4 \choose 3}{3 \choose 1}^3{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,728 || 0.0088807 ||align=right|    111.6 : 1
 * align=right|   2,160 || 0.0111008 ||align=right|     89.1 : 1
 * colspan=2| Total
 * align=right|  64,464 || 0.3312982 ||align=right|      2.0 : 1
 * align=right|  92,340 || 0.4745606 ||align=right|      1.1 : 1
 * align=right|  92,340 || 0.4745606 ||align=right|      1.1 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 2}{3 \choose 1}^2 \end{matrix}$$
 * align=right|     324 || 0.0016651 ||align=right|    599.6 : 1
 * align=right|     324 || 0.0016651 ||align=right|    599.6 : 1
 * $$\begin{matrix} {4 \choose 4}{3 \choose 1}^4 \end{matrix}$$
 * align=right|      81 || 0.0004163 ||align=right|  2,401.2 : 1
 * align=right|      81 || 0.0004163 ||align=right|  2,401.2 : 1
 * $$\begin{matrix} {4 \choose 3}{3 \choose 1}^3{52-4r \choose 1} \end{matrix}$$
 * align=right|   2,160 || 0.0111008 ||align=right|     89.1 : 1
 * align=right|   1,728 || 0.0088807 ||align=right|    111.6 : 1
 * colspan=2| Total
 * align=right|   2,565 || 0.131822 ||align=right|     74.9 : 1
 * align=right|   2,133 || 0.109621 ||align=right|     90.2 : 1
 * align=right|   2,133 || 0.109621 ||align=right|     90.2 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|       4 || 0.0000206 ||align=right| 48,644.0 : 1
 * align=right|       5 || 0.0000257 ||align=right| 38,915.0 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 3}{r-5 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     192 || 0.0009867 ||align=right|  1,012.4 : 1
 * align=right|     320 || 0.0016446 ||align=right|    607.1 : 1
 * $$\begin{matrix} {r-4 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|     216 || 0.0011101 ||align=right|    899.8 : 1
 * align=right|     360 || 0.0018501 ||align=right|    539.5 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|     320 || 0.0016446 ||align=right|    607.1 : 1
 * align=right|     320 || 0.0016446 ||align=right|    607.1 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   5,760 || 0.0296022 ||align=right|     32.8 : 1
 * align=right|   7,680 || 0.0394696 ||align=right|     24.3 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 3}{3 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|      36 || 0.0001850 ||align=right|  5,404.0 : 1
 * align=right|      36 || 0.0001850 ||align=right|  5,404.0 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 3}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      64 || 0.0003289 ||align=right|  3,039.3 : 1
 * align=right|      80 || 0.0004111 ||align=right|  2,431.3 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|      80 || 0.0004111 ||align=right|  2,431.3 : 1
 * align=right|      64 || 0.0003289 ||align=right|  3,039.3 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 2}^2 \end{matrix}$$
 * align=right|      54 || 0.0002775 ||align=right|  3,602.3 : 1
 * align=right|      54 || 0.0002775 ||align=right|  3,602.3 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   2,160 || 0.0111008 ||align=right|     89.1 : 1
 * align=right|   1,728 || 0.0088807 ||align=right|    111.6 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     270 || 0.0013876 ||align=right|    719.7 : 1
 * align=right|     288 || 0.0014801 ||align=right|    674.6 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   3,840 || 0.0197348 ||align=right|     49.7 : 1
 * align=right|   3,840 || 0.0197348 ||align=right|     49.7 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     192 || 0.0009867 ||align=right|  1,012.4 : 1
 * align=right|     240 || 0.0012334 ||align=right|    809.8 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   5,760 || 0.0296022 ||align=right|     32.8 : 1
 * align=right|   5,760 || 0.0296022 ||align=right|     32.8 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 2} \end{matrix}$$
 * align=right|  71,820 || 0.3691027 ||align=right|      1.7 : 1
 * align=right|  59,520 || 0.3058896 ||align=right|      2.3 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 3} \end{matrix}$$
 * align=right|  31,920 || 0.1640456 ||align=right|      5.1 : 1
 * align=right|  17,920 || 0.0920958 ||align=right|      9.9 : 1
 * $$\begin{matrix} {52-4r \choose 4} \end{matrix}$$
 * align=right|   4,845 || 0.0248998 ||align=right|     39.2 : 1
 * align=right|   1,820 || 0.0093535 ||align=right|    105.9 : 1
 * colspan=2| Total
 * align=right| 127,551 || 0.6555196 ||align=right|      0.5 : 1
 * align=right| 100,107 || 0.5144773 ||align=right|      0.9 : 1
 * }
 * }


 * {| border="2" cellpadding="2" cellspacing="0" style="border: 3px #aaa solid; font-size: 95%;"

! rowspan=2| Hand to make !! rowspan=2| Board !! rowspan=2| Derivation ! colspan=3| 8-high qualifier (r = 8) ! colspan=3| 9-high qualifier (r = 9) ! Combos !! Probability !! Odds !! Combos !! Probability !! Odds ! rowspan=35| Make low ! ABCDE ! AABCD ! ABCD* ! AABBC ! AABC* ! AAABC ! ABC** ! xABCD ! xABC* ! xAABC ! xxABC ! xyABC ! xxxAB ! xxyAB ! xxAAB ! xxAB* ! xyAAB ! xyAB* ! xAAAB ! xAABB ! xAAB* ! xAB** ! xyzAB ! xxxyA ! xxyyA ! xxyAA ! xxyA* ! xyAAA ! xyAA* ! xyA** ! xxyzA ! xyzAA ! xyzA* ! xyzrA ! rowspan=7| Miss low ! xxxyz ! xxyyz ! xxyz* ! xyz** ! xxyzr ! xyzr* ! rowspan=24| No low ! AAAAB ! AAAA* ! AAABB ! AAAB* ! AABB* ! AAA** ! AAB** ! xxxyy ! xxxy* ! xxxAA ! xxxA* ! xxx** ! xxyy* ! xxy** ! xxAAA ! xxAA* ! xxA** ! xAAAA ! xAAA* ! xAA** ! ??*** ! ?**** ! *****
 * +Derivations for low hand shape xyzr (four low ranks) on the river
 * - style="Background: #eee"
 * - style="Background: #eee"
 * $$\begin{matrix} {r-4 \choose 5}{4 \choose 1}^5 \end{matrix}$$ (see #1 below)
 * align=right|       0 || 0.0000000 ||align=right|  &infin; : 1
 * align=right|   1,024 || 0.0005980 ||align=right|  1,671.2 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 2}{r-5 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   1,536 || 0.0008970 ||align=right|  1,113.8 : 1
 * align=right|   7,680 || 0.0044852 ||align=right|    222.0 : 1
 * $$\begin{matrix} {r-4 \choose 4}{4 \choose 1}^4{52-4r \choose 1} \end{matrix}$$
 * align=right|   5,120 || 0.0029901 ||align=right|    333.4 : 1
 * align=right|  20,480 || 0.0119605 ||align=right|     82.6 : 1
 * $$\begin{matrix} {r-4 \choose 2}{4 \choose 2}^2{r-5 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,728 || 0.0010092 ||align=right|    989.9 : 1
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 2}{r-5 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * align=right|  46,080 || 0.0269111 ||align=right|     36.2 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 3}{r-5 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     768 || 0.0004485 ||align=right|  2,228.6 : 1
 * align=right|   1,920 || 0.0011213 ||align=right|    890.8 : 1
 * $$\begin{matrix} {r-4 \choose 3}{4 \choose 1}^3{52-4r \choose 2} \end{matrix}$$
 * align=right|  48,640 || 0.0284062 ||align=right|     34.2 : 1
 * align=right|  76,800 || 0.0448518 ||align=right|     21.3 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 4}{4 \choose 1}^4 \end{matrix}$$
 * align=right|   3,072 || 0.0017941 ||align=right|    556.4 : 1
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 3}{4 \choose 1}^3{52-4r \choose 1} \end{matrix}$$
 * align=right|  61,440 || 0.0358815 ||align=right|     26.9 : 1
 * align=right| 122,880 || 0.0717630 ||align=right|     12.9 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  13,824 || 0.0080733 ||align=right|    122.9 : 1
 * align=right|  34,560 || 0.0201833 ||align=right|     48.5 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|   3,072 || 0.0017941 ||align=right|    556.4 : 1
 * align=right|   7,680 || 0.0044852 ||align=right|    222.0 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 3}{4 \choose 1}^3 \end{matrix}$$
 * align=right|  13,824 || 0.0080733 ||align=right|    122.9 : 1
 * align=right|  34,560 || 0.0201833 ||align=right|     48.5 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 3}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|     384 || 0.0002243 ||align=right|  4,458.1 : 1
 * align=right|     640 || 0.0003738 ||align=right|  2,674.5 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  10,368 || 0.0060550 ||align=right|    164.2 : 1
 * align=right|  17,280 || 0.0100917 ||align=right|     98.1 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   3,456 || 0.0020183 ||align=right|    494.5 : 1
 * align=right|   5,760 || 0.0033639 ||align=right|    296.3 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|  23,040 || 0.0134556 ||align=right|     73.3 : 1
 * align=right|  30,720 || 0.0179407 ||align=right|     54.7 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 2}{r-5 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|  15,552 || 0.0090825 ||align=right|    109.1 : 1
 * align=right|  25,920 || 0.0151375 ||align=right|     65.1 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 2}{4 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right| 103,680 || 0.0605500 ||align=right|     15.5 : 1
 * align=right| 138,240 || 0.0807333 ||align=right|     11.4 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 3}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   2,304 || 0.0013456 ||align=right|    742.2 : 1
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 2}^2 \end{matrix}$$
 * align=right|   2,592 || 0.0015137 ||align=right|    659.6 : 1
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-1 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|  69,120 || 0.0403667 ||align=right|     23.8 : 1
 * align=right|  92,160 || 0.0538222 ||align=right|     17.6 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-1 \choose 2}{4 \choose 1}^2{52-4r \choose 2} \end{matrix}$$
 * align=right| 218,880 || 0.1278278 ||align=right|      6.8 : 1
 * align=right| 230,400 || 0.1345555 ||align=right|      6.4 : 1
 * $$\begin{matrix} {4 \choose 3}{3 \choose 1}^3{r-4 \choose 2}{4 \choose 1}^2 \end{matrix}$$
 * align=right|  10,368 || 0.0060550 ||align=right|    164.2 : 1
 * align=right|  17,280 || 0.0100917 ||align=right|     98.1 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 3}{3 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     576 || 0.0003364 ||align=right|  2,971.8 : 1
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 2}^2{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|     864 || 0.0005046 ||align=right|  1,980.8 : 1
 * align=right|   1,080 || 0.0006307 ||align=right|  1,584.5 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|   1,620 || 0.0009461 ||align=right|  1,056.0 : 1
 * align=right|   2,592 || 0.0015137 ||align=right|    659.6 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|  34,560 || 0.0201833 ||align=right|     48.5 : 1
 * align=right|  34,560 || 0.0201833 ||align=right|     48.5 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     864 || 0.0005046 ||align=right|  1,980.8 : 1
 * align=right|   1,080 || 0.0006307 ||align=right|  1,584.5 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|  25,920 || 0.0151375 ||align=right|     65.1 : 1
 * align=right|  25,920 || 0.0151375 ||align=right|     65.1 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 1}^2{r-4 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right| 164,160 || 0.0958708 ||align=right|      9.4 : 1
 * align=right| 129,600 || 0.0756875 ||align=right|     12.2 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 2}{3 \choose 1}^2 \end{matrix}$$
 * align=right|   5,184 || 0.0030275 ||align=right|    329.3 : 1
 * align=right|   6,480 || 0.0037844 ||align=right|    263.2 : 1
 * $$\begin{matrix} {4 \choose 3}{3 \choose 1}^3{r-4 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|   2,592 || 0.0015137 ||align=right|    659.6 : 1
 * align=right|   3,240 || 0.0018922 ||align=right|    527.5 : 1
 * $$\begin{matrix} {4 \choose 3}{3 \choose 1}^3{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|  34,560 || 0.0201833 ||align=right|     48.5 : 1
 * align=right|  34,560 || 0.0201833 ||align=right|     48.5 : 1
 * $$\begin{matrix} {4 \choose 4}{3 \choose 1}^4{r-4 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|   1,296 || 0.0007569 ||align=right|  1,320.2 : 1
 * align=right|   1,620 || 0.0009461 ||align=right|  1,056.0 : 1
 * colspan=2| Total
 * align=right| 908,976 || 0.5308497 ||align=right|      0.9 : 1
 * align=right|1,182,004 || 0.6903003 ||align=right|     0.4 : 1
 * align=right|1,182,004 || 0.6903003 ||align=right|     0.4 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 3}{3 \choose 2}{3 \choose 1}^2 \end{matrix}$$
 * align=right|     108 || 0.0000631 ||align=right| 15,853.7 : 1
 * align=right|     108 || 0.0000631 ||align=right| 15,853.7 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 2}^2{2 \choose 1}{3 \choose 1} \end{matrix}$$
 * align=right|     324 || 0.0001892 ||align=right|  5,283.9 : 1
 * align=right|     324 || 0.0001892 ||align=right|  5,283.9 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 2}{3 \choose 1}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|   6,480 || 0.0037844 ||align=right|    263.2 : 1
 * align=right|   5,184 || 0.0030275 ||align=right|    329.3 : 1
 * $$\begin{matrix} {4 \choose 3}{3 \choose 1}^3{52-4r \choose 2} \end{matrix}$$
 * align=right|  20,520 || 0.0119839 ||align=right|     82.4 : 1
 * align=right|  12,960 || 0.0075687 ||align=right|    131.1 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 3}{3 \choose 1}^3 \end{matrix}$$
 * align=right|     324 || 0.0001892 ||align=right|  5,283.9 : 1
 * align=right|     324 || 0.0001892 ||align=right|  5,283.9 : 1
 * $$\begin{matrix} {4 \choose 4}{3 \choose 1}^4{52-4r \choose 2} \end{matrix}$$
 * align=right|   1,620 || 0.0009461 ||align=right|  1,056.0 : 1
 * align=right|   1,296 || 0.0007569 ||align=right|  1,320.2 : 1
 * colspan=2| Total
 * align=right|  29,376 || 0.0171558 ||align=right|     57.3 : 1
 * align=right|  20,196 || 0.0117946 ||align=right|     83.8 : 1
 * align=right|  20,196 || 0.0117946 ||align=right|     83.8 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 4}{r-5 \choose 1}{4 \choose 1} \end{matrix}$$
 * align=right|      48 || 0.0000280 ||align=right| 35,672.0 : 1
 * align=right|      80 || 0.0000467 ||align=right| 21,402.8 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 4}{52-4r \choose 1} \end{matrix}$$
 * align=right|      80 || 0.0000467 ||align=right| 21,402.8 : 1
 * align=right|      80 || 0.0000467 ||align=right| 21,402.8 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 3}{r-5 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|     288 || 0.0001682 ||align=right|  5,944.5 : 1
 * align=right|     480 || 0.0002803 ||align=right|  3,566.3 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 3}{r-5 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * align=right|   5,120 || 0.0029901 ||align=right|    333.4 : 1
 * $$\begin{matrix} {r-4 \choose 2}{4 \choose 2}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|   4,320 || 0.0025229 ||align=right|    395.4 : 1
 * align=right|   5,760 || 0.0033639 ||align=right|    296.3 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 3}{52-4r \choose 2} \end{matrix}$$
 * align=right|   3,040 || 0.0017754 ||align=right|    562.3 : 1
 * align=right|   2,400 || 0.0014016 ||align=right|    712.5 : 1
 * $$\begin{matrix} {r-4 \choose 1}{4 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|  54,720 || 0.0319569 ||align=right|     30.3 : 1
 * align=right|  57,600 || 0.0336389 ||align=right|     28.7 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 3}{3 \choose 1}{3 \choose 2} \end{matrix}$$
 * align=right|      36 || 0.0000210 ||align=right| 47,563.0 : 1
 * align=right|      36 || 0.0000210 ||align=right| 47,563.0 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 3}{3 \choose 1}{3 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|     720 || 0.0004205 ||align=right|  2,377.2 : 1
 * align=right|     576 || 0.0003364 ||align=right|  2,971.8 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 3}{r-4 \choose 1}{4 \choose 2} \end{matrix}$$
 * align=right|      96 || 0.0000561 ||align=right| 17,835.5 : 1
 * align=right|     120 || 0.0000701 ||align=right| 14,268.2 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 3}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 1} \end{matrix}$$
 * align=right|   1,280 || 0.0007475 ||align=right|  1,336.7 : 1
 * align=right|   1,280 || 0.0007475 ||align=right|  1,336.7 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 3}{52-4r \choose 2} \end{matrix}$$
 * align=right|     760 || 0.0004438 ||align=right|  2,252.0 : 1
 * align=right|     480 || 0.0002803 ||align=right|  3,566.3 : 1
 * $$\begin{matrix} {4 \choose 2}{3 \choose 2}^2{52-4r \choose 1} \end{matrix}$$
 * align=right|   1,080 || 0.0006307 ||align=right|  1,584.5 : 1
 * align=right|     864 || 0.0005046 ||align=right|  1,980.8 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{3 \choose 1}{3 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|  20,520 || 0.0119839 ||align=right|     82.4 : 1
 * align=right|  12,960 || 0.0075687 ||align=right|    131.1 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 3} \end{matrix}$$
 * align=right|     192 || 0.0001121 ||align=right|  8,917.3 : 1
 * align=right|     240 || 0.0001402 ||align=right|  7,133.6 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 2}{52-4r \choose 1} \end{matrix}$$
 * align=right|   5,760 || 0.0033639 ||align=right|    296.3 : 1
 * align=right|   5,760 || 0.0033639 ||align=right|    296.3 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 2}{r-4 \choose 1}{4 \choose 1}{52-4r \choose 2} \end{matrix}$$
 * align=right|  36,480 || 0.0213046 ||align=right|     45.9 : 1
 * align=right|  28,800 || 0.0168194 ||align=right|     58.5 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 4} \end{matrix}$$
 * align=right|      48 || 0.0000280 ||align=right| 35,672.0 : 1
 * align=right|      60 || 0.0000350 ||align=right| 28,537.4 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 3}{52-4r \choose 1} \end{matrix}$$
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * align=right|   3,840 || 0.0022426 ||align=right|    444.9 : 1
 * $$\begin{matrix} {4 \choose 1}{3 \choose 1}{r-4 \choose 1}{4 \choose 2}{52-4r \choose 2} \end{matrix}$$
 * align=right|  54,720 || 0.0319569 ||align=right|     30.3 : 1
 * align=right|  43,200 || 0.0252292 ||align=right|     38.6 : 1
 * $$\begin{matrix} {4r-4 \choose 2}{52-4r \choose 3} \end{matrix}$$
 * align=right| 430,920 || 0.2516609 ||align=right|      3.0 : 1
 * align=right| 277,760 || 0.1622142 ||align=right|      5.2 : 1
 * $$\begin{matrix} {4r-4 \choose 1}{52-4r \choose 4} \end{matrix}$$
 * align=right| 135,660 || 0.0792266 ||align=right|     11.6 : 1
 * align=right|  58,240 || 0.0340127 ||align=right|     28.4 : 1
 * $$\begin{matrix} {52-4r \choose 5} \end{matrix}$$
 * align=right|  15,504 || 0.0090545 ||align=right|    109.4 : 1
 * align=right|   4,368 || 0.0025509 ||align=right|    391.0 : 1
 * colspan=2| Total
 * align=right| 773,952 || 0.4519945 ||align=right|      1.2 : 1
 * align=right| 510,104 || 0.2979050 ||align=right|      2.4 : 1
 * }
 * 1) The board ABCDE is not possible with an 8-high qualifier since there are only 8 − 4 = 4 low ranks not represented in the hand xyzr.  In this case, by definition the binomial coefficient $$\begin{matrix} {8-5 \choose 4} = 0 \end{matrix}.$$
 * 1) The board ABCDE is not possible with an 8-high qualifier since there are only 8 − 4 = 4 low ranks not represented in the hand xyzr.  In this case, by definition the binomial coefficient $$\begin{matrix} {8-5 \choose 4} = 0 \end{matrix}.$$