A closet contains n pairs of shoes. (a) if 2r shoes are chosen at random (2r < n), what is the probability that there will be no matching pair in the sample?

Answer: 2^ (2r) C (2r, n) / C (2r, 2n)

Solution: The number of ways to select 2r out of 2n shoes is as follows:

C (2r, 2n)

Now suppose that n pairs of shoes are (a1, b1), (a2, b2), ..., (an, bn).

When we select 2r in here, there is no matching pair, so we can choose at most one in each pair. The number of ways for selecting 2r pairs in n pairs is as follows:

C (2r, n)

Since we can choose one of the two in a pair, the number of ways we can pull it is as follows

C (2r, n) * 2^ (2r)

Therefore, the probabilty of getting is:

C (2r, n) * 2^ (2r) / C (2r, 2n)