A permutation of $\{1,2, \ldots, n\}$ is drawn at random. What is the average number of fixed points
Let $X$ be the set of pairs $(x, \sigma)$ where $x$ is a fixed point of the permutation $\sigma$. For every $x$, there are $(n-1)!$ permutations that fix $x$, so $|X|=n \times(n-1)!=n!$

On the other hand, we have $|X|=\sum_{\sigma}|Fix(\sigma)|$, so the average number of fixed points is equal to 1.