Example 2-56 In the permutations of the 8 letters A, B, C, D, E, F, G, H, find the number of derangements where the letters A, C, E, G are not in their original positions.
In the full permutation of 8 letters, let $A_{1}, A_{2}, A_{3}, A_{4}$ represent the permutations where $\mathrm{A}, \mathrm{C}, \mathrm{E}, \mathrm{G}$ are in their original positions, then
$$
\begin{aligned}
\left|\bar{A}_{1} \cap \bar{A}_{2} \cap \bar{A}_{3} \cap \bar{A}_{4}\right| & =8!-C(4,1) 7!+C(4,2) 6!-C(4,3) 5!+C(4,4) 4! \\
& =40320-20160+4320-480+24=24024
\end{aligned}
$$