State the inclusion-exclusion principle.

Let $n \in \mathbb{N}$. A permutation $\sigma$ of the set $\{1,2,3, \ldots, n\}$ is said to contain a transposition if there exist $i, j$ with $1 \leqslant i<j \leqslant n$ such that $\sigma(i)=j$ and $\sigma(j)=i$. Derive a formula for the number, $f(n)$, of permutations which do not contain a transposition, and show that

$$\lim _{n \rightarrow \infty} \frac{f(n)}{n !}=e^{-\frac{1}{2}}$$