Probability | Part IA, 2003

State the inclusion-exclusion formula for the probability that at least one of the events A1,A2,,AnA_{1}, A_{2}, \ldots, A_{n} occurs.

After a party the nn guests take coats randomly from a pile of their nn coats. Calculate the probability that no-one goes home with the correct coat.

Let p(m,n)p(m, n) be the probability that exactly mm guests go home with the correct coats. By relating p(m,n)p(m, n) to p(0,nm)p(0, n-m), or otherwise, determine p(m,n)p(m, n) and deduce that

limnp(m,n)=1em!.\lim _{n \rightarrow \infty} p(m, n)=\frac{1}{e m !} .

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