Wikipedia:Category combinatorics

This page illustrates how some categories of articles can be categorized in larger categories by means of combinatorics, binomial expansion, and Pascal's triangle. The section "One factor" is included for continuation of the pattern.

The pattern can be complicated by intermediate categories with names having phrases such as "by nationality" (or "by country"), "by religious status", "by sex", "by occupation", "by employment status", and "by marital status".
 * Category:American scientists ⊆ Category:Scientists by nationality ⊆ Category:Scientists
 * Category:American scientists ⊆ Category:American people by occupation ⊆ Category:Americans.

The guidelines at Category names might not apply universal consistency to the order of elements in category names.

One element in a category name

 * Combination (1) of 1 element
 * X: Category:Scientists


 * Combination (1) of 0 element
 * _: Category:Contents


 * 1 + 1 = 2

Two elements in a category name

 * Combination (1) of 2 elements
 * XX: Category:American scientists


 * Combinations (2) of 1 element
 * X_: Category:Americans
 * _X: Category:Scientists


 * Combination (1) of 0 element
 * __: Category:Contents


 * 1 + 2 + 1 = 4 = 2 × 2

Three elements in a category name

 * Combination (1) of 3 elements
 * XXX: Category:American women scientists


 * Combinations (3) of 2 elements
 * XX_: Category:American women
 * X_X: Category:American scientists
 * _XX: Category:Women scientists


 * Combinations (3) of 1 element
 * X__: Category:Americans
 * _X_: Category:Women
 * __X: Category:Scientists


 * Combination (1) of 0 element
 * ___: Category:Contents


 * 1 + 3 + 3 + 1 = 8 = 2 × 2 × 2

Four elements in a category name

 * Combination (1) of 4 elements
 * XXXX: Category:American Jewish women scientists


 * Combinations (4) of 3 elements
 * XXX_: Category:American Jewish women
 * XX_X: Category:American Jewish scientists
 * X_XX: Category:American women scientists
 * _XXX: Category:Jewish women scientists


 * Combinations (6) of 2 elements
 * XX__: Category:American Jews
 * X_X_: Category:American women
 * X__X: Category:American scientists
 * _XX_: Category:Jewish women
 * _X_X: Category:Jewish scientists
 * __XX: Category:Women scientists


 * Combinations (4) of 1 element
 * X___: Category:Americans
 * _X__: Category:Jews
 * __X_: Category:Women
 * ___X: Category:Scientists


 * Combination (1) of 0 element
 * ____: Category:Contents


 * 1 + 4 + 6 + 4 + 1 = 16 = 2 × 2 × 2 × 2

Five elements in a category name

 * Combination (1) of 5 elements
 * XXXXX: Category:Lists of American Jewish women scientists


 * Combinations (5) of 4 elements
 * XXXX_: Category:Lists of American Jewish women
 * XXX_X: Category:Lists of American Jewish scientists
 * XX_XX: Category:Lists of American women scientists
 * X_XXX: Category:Lists of Jewish women scientists
 * _XXXX: Category:American Jewish women scientists


 * Combinations (10) of 3 elements
 * XXX__: Category:Lists of American Jews
 * XX_X_: Category:Lists of American women
 * XX__X: Category:Lists of American scientists
 * X_XX_: Category:Lists of Jewish women
 * X_X_X: Category:Lists of Jewish scientists
 * X__XX: Category:Lists of women scientists
 * _XXX_: Category:American Jewish women
 * _XX_X: Category:American Jewish scientists
 * _X_XX: Category:American women scientists
 * __XXX: Category:Jewish women scientists


 * Combinations (10) of 2 elements
 * XX___: Category:Lists of Americans
 * X_X__: Category:Lists of Jews
 * X__X_: Category:Lists of women
 * X___X: Category:Lists of scientists
 * _XX__: Category:American Jews
 * _X_X_: Category:American women
 * _X__X: Category:American scientists
 * __XX_: Category:Jewish women
 * __X_X: Category:Jewish scientists
 * ___XX: Category:Women scientists


 * Combinations (5) of 1 element
 * X____: Category:Lists
 * _X___: Category:Americans
 * __X__: Category:Jews
 * ___X_: Category:Women
 * ____X: Category:Scientists


 * Combination (1) of 0 element
 * _____: Category:Contents


 * 1 + 5 + 10 + 10 + 5 + 1 = 32 = 2 × 2 × 2 × 2 × 2

Six elements in a category name

 * Combination (1) of 6 elements
 * XXXXXX: Category:Lists of award-winning American Jewish women scientists


 * Combinations (6) of 5 elements
 * XXXXX_: Category:Lists of award-winning American Jewish women
 * XXXX_X: Category:Lists of award-winning American Jewish scientists
 * XXX_XX: Category:Lists of award-winning American women scientists
 * XX_XXX: Category:Lists of award-winning Jewish women scientists
 * X_XXXX: Category:Lists of American Jewish women scientists
 * _XXXXX: Category:Award-winning American Jewish women scientists


 * Combinations (15) of 4 elements
 * XXXX__: Category:Lists of award-winning American Jews
 * XXX_X_: Category:Lists of award-winning American women
 * XXX__X: Category:Lists of award-winning American scientists
 * XX_XX_: Category:Lists of award-winning Jewish women
 * XX_X_X: Category:Lists of award-winning Jewish scientists
 * XX__XX: Category:Lists of award-winning women scientists
 * X_XXX_: Category:Lists of American Jewish women
 * X_XX_X: Category:Lists of American Jewish scientists
 * X_X_XX: Category:Lists of American women scientists
 * X__XXX: Category:Lists of Jewish women scientists
 * _XXXX_: Category:Award-winning American Jewish women
 * _XXX_X: Category:Award-winning American Jewish scientists
 * _XX_XX: Category:Award-winning Americanh women scientists
 * _X_XXX: Category:Award-winning Jewish women scientists
 * __XXXX: Category:American Jewish women scientists


 * Combinations (20) of 3 elements
 * XXX___: Category:Lists of award-winning Americans
 * XX_X__: Category:Lists of award-winning Jews
 * XX__X_: Category:Lists of award-winning women
 * XX___X: Category:Lists of award-winning scientists
 * X_XX__: Category:Lists of American Jews
 * X_X_X_: Category:Lists of American women
 * X_X__X: Category:Lists of American scientists
 * X__XX_: Category:Lists of Jewish women
 * X__X_X: Category:Lists of Jewish scientists
 * X___XX: Category:Lists of women scientists
 * _XXX__: Category:Award-winning American Jewish
 * _XX_X_: Category:Award-winning American women
 * _XX__X: Category:Award-winning American scientists
 * _X_XX_: Category:Award-winning Jewish women
 * _X_X_X: Category:Award-winning Jewish scientists
 * _X__XX: Category:Award-winning women scientists
 * __XXX_: Category:American Jewish women
 * __XX_X: Category:American Jewish scientists
 * __X_XX: Category:American women scientists
 * ___XXX: Category:Jewish women scientists


 * Combinations (15) of 2 elements
 * XX____: Category:Lists of award-winners
 * X_X___: Category:Lists Americans
 * X__X__: Category:Lists of Jews
 * X___X_: Category:Lists of women
 * X____X: Category:Lists of scientists
 * _XX___: Category:Award-winning Americans
 * _X_X__: Category:Award-winning Jews
 * _X__X_: Category:Award-winning women
 * _X___X: Category:Award-winning scientists
 * __XX__: Category:American Jews
 * __X_X_: Category:American women
 * __X__X: Category:American scientists
 * ___XX_: Category:Jewish women
 * ___X_X: Category:Jewish scientists
 * ____XX: Category:Women scientists


 * Combination (6) of 1 element
 * X_____: Category:Lists
 * _X____: Category:Award-winners
 * __X___: Category:Americans
 * ___X__: Category:Jews
 * ____X_: Category:Women
 * _____X: Category:Scientists


 * Combination (1) of 0 element
 * ______: Category:Contents


 * 1 + 6 + 15 + 20 + 15 + 6 + 1 = 64 = 2 × 2 × 2 × 2 × 2 × 2