S Fixed points of a random permutation. Before a theatre performance, n people leave their coats in the cloak room. After the performance, due to a power cut, the coats are returned in the dark in random order. Let X be the random number of people who get their own coat back. Find the distribution of X, i.e., P.X D k/ for every k  0. What happens in the limit as n!1? (The case k D 0 corresponds to the rencontre problem of Problem 1.13.) Hint:

Use Problem 1.7 again.