The well-known "hats" problem, in which a number of people enter a restaurant and check their hats, and then receive them back at random, is often used to illustrate the concept of derangements, that is, permutations with no fixed points. In this paper, the problem is extended to multiple items of clothing, and a general solution to the problem of multiple coincidences is obtained using the technique of binomial inversion. This solution also motivates a general formula for the expected number of people who return home with all of their own clothes.