Exactly one of five people draws their own name

Question

Five people draw names from a hat, giving a uniformly random assignment of the five names (a random permutation), all $5! = 120$ equally likely.

What is the probability that exactly one person draws their own name?

Accepted Answer

"Exactly one match" means one person is matched and the other four are all mismatched. Build such an arrangement in two independent steps: 1. Choose which single person is the lucky match: \binom{5}{1} = 5 ways. 2. The remaining four names must be a derangement of the other four people, so nobody else matches: D4 = 9 ways. So the number of arrangements with exactly one match is 5 	imes 9 = 45, and P(	ext{exactly one match}) = \frac{5 \cdot D4}{5!} = \frac{45}{120} = \frac{3}{8} = 0.375. Takeaway. For exactly one fixed point among n, pick the matcher (n ways) and derange the rest (D{n-1}), g…

Exactly one of five people draws their own name

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