41. Let $\pi=a_{1} a_{2} \cdots a_{n}$ be any permutation of $1,2, \cdots, n$. If $a_{i}=$ $i(1 \leqslant i \leqslant n)$, then $i$ is said to be in its position in the permutation $\pi$. Let $D_{n}$ denote the number of permutations of $1,2, \cdots, n$ in which no number is in its position. Find the counting formula for $D_{n}$.
41. $D_{n}=n!\sum_{k=0}^{n}(-1)^{k} / k!$.