A few days ago, Amy asked me an interesting question. Say you have a deck of N cards, n1 of one type, n2 of the second type, and so on, and you shuffle them all together. Then you predict the order in which you will see all of the cards, and flip them over one-by-one. How many will you guess right, on average?

This problem turned out to be tricker than I thought it would be, and since it doesn’t show up on random internet searches, I figured it would be good to work through the problem here. It’s related to the problem of counting derangements (permutations of N elements with no fixed point), which was solved by Bernoulli in the early 1700’s; but the version with groups of identical objects turns out to be slightly messier. In the same way that multinomials generalize binomials, these “multinomial derangements” generalize ordinary derangements.

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