Exactly two of five players draw their own jersey

Question

Five players receive their five jerseys back in a completely random order, so all 5! = 120 assignments are equally likely. What is the probability that exactly two players get their own jersey?

Accepted Answer

Exactly two matches means two players are matched and the other three are all mismatched. Count in two steps: 1. Choose which two players are matched: \binom{5}{2} = 10 ways. 2. Derange the remaining three jerseys so nobody else matches: D3 = 2 ways. So there are 10 	imes 2 = 20 favorable arrangements, and P(	ext{exactly two matches}) = \frac{\binom{5}{2}\,D3}{5!} = \frac{20}{120} = \frac{1}{6} \approx 0.1667. Takeaway. For exactly m fixed points, choose the m matchers (\binom{n}{m} ways) and derange the remaining n-m, giving \binom{n}{m}D{n-m}/n!.

Exactly two of five players draw their own jersey

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