N guests arrive at a party. Each person is wearing a hat. We collect all the hats and then randomly redistribute the hats, giving each person one of the N hats randomly. Let X_N be the number of people who receive their own hats. Find the PMF of X_N.

We previously found that (Problem 7 in Section 2.1.5)

Problem 7 in Section 2.1.5

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P(XN = 0) - 1 11 + 3! 4! 2! • (-1)^_1 N! for N = 1,2,.... Using this, find P(XN = k) for all k = {0, 1,... N}.