Exercise $2\mathrm{.}31.(*)$

This question has three parts.

$($a$)$ The subfactorial of a number, namely $!n,$ is the number of permutations of $n$ objects such that no object appears in its natural spot. For example, take the collection of objects $\{a,b,c\}.$ There are $6$ possible permutations $($because we choose arrangements for three items, and $36,$ but only two of th