Example 20: A person writes $n$ letters to $n$ people, one for each, and prepares $n$ envelopes with the names and addresses of the recipients written on them. How many different ways are there to place the letters into the envelopes (one letter per envelope) so that no letter is placed in the envelope intended for its recipient?
Let the set of all different ways to place the letters be $S$, where $A_{i}$ represents the set of methods in which the $i$-th letter is exactly placed in the envelope with the name and address of the $i$-th person $(i=1,2,3, \cdots, n)$. Thus,
$$
\begin{array}{l}
|S|=n!,\left|A_{i}\right|=(n-1)!(1 \leqslant i \leqslant n),\left|A_{i} \cap A_{j}\right|=(n-2)!(1 \leqslant i<j \leqslant \\
n), \cdots,\left|A_{i_{1}} \cap A_{i_{2}} \cap \cdots \cap A_{i_{k}}\right|=(n-k)!\left(1 \leqslant i_{1}<i_{2}<\cdots<i_{k} \leqslant n\right), \\
\left|A_{1} \cap A_{2} \cap \cdots \cap A_{n}\right|=1 .
\end{array}
$$

By the principle of inclusion-exclusion, the number of methods sought is
$$
\begin{aligned}
D_{n}= & \left|\bar{A}_{1} \cap \bar{A}_{2} \cap \cdots \cap \bar{A}_{n}\right|=|S|-\sum_{i=1}^{n}\left|A_{i}\right|+\sum_{1 \leqslant i<j \leqslant n}\left|A_{i} \cap A_{j}\right|- \\
& \sum_{1 \leqslant j<j<k \leqslant n}\left|A_{i} \cap A_{j} \cap A_{k}\right|+\cdots+(-1)^{n}\left|A_{1} \cap A_{2} \cap \cdots \cap A_{n}\right| \\
= & n!-C_{n}^{1} \cdot(n-1)!+C_{n}^{2}(n-2)!-C_{n}^{3}(n-3)!+\cdots+(-1)^{n} C_{n}^{n} \cdot 1 \\
= & n!\left[1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots+\frac{(-1)^{n}}{n!}\right] .
\end{aligned}
$$