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

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

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

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