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You have n balls, and n boxes. There is a pairing of each ball to a box (and vice versa). If you were to randomly place balls in boxes, what is the probability that none of the balls would go in "their" box?
I've solved this numerically for:
n | probability --------------- 1 | 0 2 | 1 / 2 3 | 1 / 3
I believe an analytic solution is possible, but I've yet to be able to formulate it.