$5 \cdot 39$ A person wrote a letter to each of 6 different recipients and prepared 6 envelopes with the recipients' names and addresses. How many ways are there to put the letters into the envelopes so that no letter matches the recipient on the envelope?
[Solution] Obviously, the total number of different ways to place the specified $i$ letters into the correct envelopes is $(6 - i)!$. Therefore, the total number of ways to place at least $i$ letters into the correct envelopes is
$$
m_{i}=C_{6}^{i}(6-i)!=\frac{6!}{i!}, \quad i=0,1, \cdots, 6,
$$

which includes repeated counting. Thus, by the principle of inclusion-exclusion, the total number of different ways to place all 6 letters into the wrong envelopes is
$$
\begin{aligned}
m & =m_{0}-m_{1}+m_{2}-m_{3}+m_{4}-m_{5}+m_{6} \\
& =6!\left(\frac{1}{0!}-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\frac{1}{4!}-\frac{1}{5!}+\frac{1}{6!}\right) \\
& =720 \times\left(\frac{1}{2}-\frac{1}{6}+\frac{1}{24}-\frac{1}{120}+\frac{1}{720}\right) \\
& =265
\end{aligned}
$$