Exactly two of six students get their own locker

Question

Six students are reassigned their six lockers in a completely random order, so all 6! = 720 assignments are equally likely. What is the probability that exactly two students get their own locker?

Accepted Answer

Exactly two matches means two students are matched and the other four are all mismatched. Count in two steps: 1. Choose which two students are matched: \binom{6}{2} = 15 ways. 2. Derange the remaining four lockers so nobody else matches: D4 = 9 ways. So there are 15 	imes 9 = 135 favorable arrangements, and P(	ext{exactly two matches}) = \frac{\binom{6}{2}\,D4}{6!} = \frac{135}{720} = \frac{3}{16} = 0.1875. Takeaway. The formula \binom{n}{m}D{n-m}/n! handles any exact match count; here m=2, n=6 gives 3/16.

Exactly two of six students get their own locker

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