Problems

A drawer contains \(n\) pairs of socks. The two socks in each pair are indistinguishable, but each pair of socks is a different colour from all the others. A set of \(2k\) socks, where \(k\) is an integer with \(2k \leqslant n\), is selected at random from this drawer: that is, every possible set of \(2k\) socks is equally likely to be selected.

1. Find the probability that, among the socks selected, there is no pair of socks.
2. Let \(X_{n,k}\) be the random variable whose value is the number of pairs of socks found amongst those selected. Show that \[\mathrm{P}(X_{n,k} = r) = \frac{\dbinom{n}{r}\dbinom{n-r}{2(k-r)}\, 2^{2(k-r)}}{\dbinom{2n}{2k}}\] for \(0 \leqslant r \leqslant k\).
3. Show that \[r\,\mathrm{P}(X_{n,k} = r) = \frac{k(2k-1)}{2n-1}\,\mathrm{P}(X_{n-1,k-1} = r-1)\,,\] for \(1 \leqslant r \leqslant k\), and hence find \(\mathrm{E}(X_{n,k})\).