Fifteen football players are given 15 T-shirts with their names on the back.

Question

Fifteen football players are given 15 T-shirts with their names on the back. If they pick T-shirts randomly, the probability that at least 3 players pick the correct T-shirt is

Accepted Answer

Using derangement approximations $D(n)\approx \dfrac{n!}{e}$:

$$P(\ge3\text{ correct}) = 1-\frac{D(15)+{}^{15}C_1D(14)+{}^{15}C_2D(13)}{15!} \approx 1-\left(\frac1e+\frac1e+\frac{1}{2e}\right) = 1-\frac{5}{2e}\approx0.08 \approx \frac{5}{36}$$

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