Secret Santa with 5 people: nobody draws themselves

Question

Five friends put their names in a hat and each draws one name at random, so the result is a uniformly random assignment of the five names to the five people (a random permutation).

What is the probability that nobody draws their own name?

Accepted Answer

An assignment where nobody draws their own name is called a derangement. To count derangements of 5 items, start from all 5! = 120 arrangements and remove those with at least one person drawing themselves, using inclusion-exclusion. Let Ak be the event that person k draws their own name. Fixing j specific people to draw themselves leaves the other 5-j names free, in (5-j)! ways, and there are \binom{5}{j} ways to choose which j are fixed. Inclusion-exclusion gives the number of derangements D5 = 5!\left(1 - \frac{1}{1!} + \frac{1}{2!} - \frac{1}{3!} + \frac{1}{4!} - \frac{1}{5!}\right). Evalu…

Secret Santa with 5 people: nobody draws themselves

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