Talk:Combinations and permutations

Moved permutations on top since they are considered more "basic". Changed the formulae n! to the more specific n!/(n-r)! and some smaller text rewrites. I'm a little slow (in my brain) using the "Show preview" button so excuse my many smaller edits... Thechamelon 20:04, 27 Apr 2005 (UTC)

Permutations with and without repetitions
The terms "permutations with repetion" and "permutations without repetition" seem inappropriate because a permutation by definition is a one-to-one and onto function $$f: S \to S$$.

So a permutation with repetition is a contradiction and a permutation without repetition is a tautology. I don't know what the proper terms should be. I'll leave the changes for someone who actually knows what to do. The problem was noticed in a discussion in alt.algebra.help; here's the relevant thread.


 * Does anyone know what would then be the proper term for a set of elements in which order matters and repetition exists? 02:32, 29 April 2006 (UTC)

+++++++++++++


 * To capitalist: the rant below also answers your question, I hope. But I start a bit earlier.

The small section titled Repetition is vague. It says: "...variants where some objects appear more than once (that is, they have some repetition)". The intention here is, probably, correct: some objects appear more than once in the original set, not in the permutations or combinations. The second sentence, about apples, is totally intractable. Should be redone.

Now, the biggest problem is in formula below, for permutation with repetitions. In this formula, repetitions are understood differently: one can repeatedly draw the same object from the original set. It means that the result of the drawing is not a subset of the original set.

What this formula refer to as permutation with repetitions is called variation:
 * http://66.102.7.104/search?q=cache:DHD2DNy2HcsJ:www.stat.ucdavis.edu/~nello/sta32/background.html+%22elementary++combinatorics%22+arrangements+combinations+permutations+&hl=en&gl=us&ct=clnk&cd=8
 * http://www.google.com/search?q=cache:KDfwx-SfOBQJ:math.uc.edu/~brycw/probab/books/smplbook/appendix/node3.html+%22elementary++combinatorics%22+variations+combinations+permutations+&hl=en&gl=us&ct=clnk&cd=1

The formula for permutation with repetitions in its traditional sense is


 * N!/(m1!m2!...mLast!),

where N = m1 + m2 + ... + mLast is cardinality of the set (total number of objects), and m1, m2, ... are numbers of identical objects of type 1, 2, ...

The formula for combination with repetitions also understands repetitions in the same wrong way. However the formula itself is now correct, it just corresponds to the different situation:
 * http://64.233.167.104/search?q=cache:oTY2-ukqDn8J:www.cs.utk.edu/~booth/311-04/notes/combinatorics.html+combinations+with+repetitions&hl=en&gl=us&ct=clnk&cd=1

Thus, the wording for combination with repetitions should be changed.

Alex -- talk to me 00:40, 26 May 2006 (UTC)

Proof
Is there any way to proove the identity Given N objects and k holes and order does not matter i.e. N1N2..Nk is the same in any order prove that the number of possible ways to do th na modda