Let $\mathbb A := \{A_1,\dots,A_n\}$ be a collection of finite sets, and for $J\subseteq[n]$ let $A_J:=\bigcap_{j\in J} A_j$. Inclusion-exclusion principle gives the number of elements belonging to at least one set from $\mathbb A$ as $$\bigg|\bigcup_{j\in[n]} A_j\bigg| = \sum_{\emptyset\ne J\subseteq [n]} (-1)^{|J|-1} |A_J|.$$ It can further be shown that the number of elements belonging to exactly one set from $\mathbb A$ can be expressed as $$\sum_{J\subseteq [n]} (-1)^{|J|-1} |A_J|\cdot |J|.$$ More generally, the number of elements belonging to exactly $k$ sets from $\mathbb A$ is given by $$\sum_{J\subseteq [n]} (-1)^{|J|-k} |A_J|\cdot \binom{|J|}k.$$
So, for functions $f(x)=(-1)^k\binom{x}{k}$ we have nice interpretations for the expression: $$\sum_{J\subseteq [n]} (-1)^{|J|} |A_J|\cdot f(|J|).$$
Q: Are there any other functions $f$, for which the expression above has a nice combinatorial interpretation?
