Example 1. The nine-digit number in the form of $1234 \times \times \times \times \times$ composed of $1,2,3,4,5,6,7,8,9$, where the ten-thousands place is not 5, the thousands place is not 6, the hundreds place is not 7, the tens place is not 8, and the units place is not 9. How many such nine-digit numbers are there?
Solution: Obviously, this is a problem of five letters being placed in the wrong envelopes. And
$$
\begin{aligned}
\overline{5} & =5!-C_{5}^{1} 4!+C_{5}^{2} 3!-C_{5}^{3} 2! \\
& +C_{5}^{4} 1!-C_{5}^{5} 0! \\
& =120-120+60-20+5-1 \\
& =44 .
\end{aligned}
$$

The number of derangements it is asking for is 44.