Hi.

I have the following combinatoric problem (well it's actually a probability problem that need to be resolved using combinatorics):

There are $\displaystyle n$ pairs of shoes in the closet.

$\displaystyle 2m$ shoes are chosen from it randomly. ($\displaystyle m<n$)

find the probability to get exactly $\displaystyle k$ pairs.

so this is what I'm thinking:

first of all: $\displaystyle |\Omega |= \binom{2n}{2m}$ (the number of possibilities to choose 2m shoes from 2n).

but I just can't decide how many possibilities are there to choose exactly $\displaystyle k$ pairs out of $\displaystyle n$.

at first I thought it would be $\displaystyle \frac{\binom{n}{k}}{\binom{2n}{2m}}$, but that's not the right answer...

any help would be greatly appretiated.

*edit

btw, acoording to what I found somwehre online, the answer is $\displaystyle \frac{\binom{n}{k}\binom{n-k}{2m-2k}2^{2m-2k}}{\binom{2n}{2m}}$

I just couldn't figure out why...