Combinatorics - choosing exactly k pairs from n

Hi.

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

There are pairs of shoes in the closet.

shoes are chosen from it randomly. ( )

find the probability to get exactly pairs.

so this is what I'm thinking:

first of all: (the number of possibilities to choose 2m shoes from 2n).

but I just can't decide how many possibilities are there to choose exactly pairs out of .

at first I thought it would be , 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

I just couldn't figure out why...

Re: Combinatorics - choosing exactly k pairs from n

Quote:

Originally Posted by

**Stormey** There are

pairs of shoes in the closet.

shoes are chosen from it randomly. (

)

find the probability to get exactly

pairs.

*edit

btw, acoording to what I found somwehre online, the answer is

I just couldn't figure out why...

I have a minor disagreement with the online answer you found.

I think it should be

Here are my reasons. Suppose we want to choose of those shoes so that **none match** (no pairs).

Clearly . There are ways to choose pairs from the pairs

From those pairs there are ways to choose either the left or the right shoe from each pair.

Does that help?

Re: Combinatorics - choosing exactly k pairs from n

Quote:

Originally Posted by

**Plato** Here are my reasons. Suppose we want to choose

of those shoes so that

**none match** (no pairs).

Clearly

. There are

ways to choose

pairs from the

pairs

From those

pairs there are

ways to choose either the left or the right shoe from each pair.

Does that help?

Actually, that makes perfect sense to me, but I still don't see why is it .

*edit

OK, now I see it.

I didn't notice that we actually choose **m** pairs of shoes, not just **k**.

Thank you very much my friend.

that was very helpful.