Example 1.31 Let $C_{n, m}$ denote the number of $m$-covers of an $n$-element set $A$. Find the counting formula for $C_{n, n}$.
Let $W$ denote the set of all $m$-element subsets of $P^{*}(A)$. Since $\left|P^{*}(A)\right|=2^{n}-1$, we have $|W|=\binom{2^{n}-1}{m}$. Now, we classify the elements of $W$ into $n$ categories as follows: Let $\Gamma \in W$. If $\left|\bigcup_{F \in \Gamma} F\right|=j(1 \leqslant j \leqslant n)$, then $\Gamma$ is called a $j$-th category element of $W$. It is clear that $\Gamma$ is a $j$-th category element of $W$ if and only if $\Gamma$ is an $m$-cover of some $j$-element subset of $A$. Since $A$ has $\binom{n}{j}$ $j$-element subsets, and a $j$-element set has $C_{j, m}$ $m$-covers, by the multiplication principle, the number of $j$-th category elements of $W$ is $\binom{n}{j} C_{j, m}$. By the addition principle, we have
$$
\binom{2^{n}-1}{m}=\sum_{j=1}^{n}\binom{n}{j} C_{j, m} .
$$

By the binomial inversion formula, we get
$$
C_{n, m}=\sum_{j=1}^{n}(-1)^{n-j}\binom{n}{j}\binom{2^{j}-1}{m} .
$$