State and prove the Inclusion-Exclusion Principle.

A permutation $\sigma$ of $\{1,2, \ldots, n\}$ is called a derangement if $\sigma(j) \neq j$ for every $j \leqslant n$. Use the Inclusion-Exclusion Principle to find a formula for the number $f(n)$ of derangements of $\{1,2, \ldots, n\}$. Show also that $f(n) / n$ ! converges to $1 / e$ as $n \rightarrow \infty$.