A closet contains $n$ different matching pairs of shoes. If $2r$ individual shoes are chosen at random ($2r < n$), what is the probability that there will be no matching pairs of shoes in the sample?
From what I understand the sample space will be, $${2n\choose 2r}$$ but I can't get my head around the number of 'successes'. Please could someone give me a hand with the thought process?
Best Answer
You need to choose the $2r$ shoes from $n$ different pairs. So there are $\binom{n}{2r}$ ways to pick the different pairs, and once that is done, there are $2^{2r}$ ways to pick one shoe from all those pairs.
So the number of "good" cases is $\binom{n}{2r}\cdot 2^{2r}$.