(a) State and prove the inclusion-exclusion formula.

(b) Let $k$ and $m$ be positive integers, let $n=k m$, let $A_{1}, \ldots, A_{k}$ be disjoint sets of size $m$, and let $A=A_{1} \cup \ldots \cup A_{k}$. Let $\mathcal{B}$ be the collection of all subsets $B \subset A$ with the following two properties:

(i) $|B|=k$;

(ii) there is at least one $i$ such that $\left|B \cap A_{i}\right|=3$.

Prove that the number of sets in $\mathcal{B}$ is given by the formula

$$\sum_{r=1}^{\lfloor k / 3\rfloor}(-1)^{r-1}\left(\begin{array}{c} k \\ r \end{array}\right)\left(\begin{array}{c} m \\ 3 \end{array}\right)^{r}\left(\begin{array}{c} n-r m \\ k-3 r \end{array}\right)$$