5. (A simple generalization of the principle of inclusion-exclusion)

Let $A$ be a finite set, and $\boldsymbol{s}=\left\{P_{1}, P_{2}, \cdots, P_{m}\right\}$ be a finite set of $m$ properties. Let $A_{i}(1 \leqslant i \leqslant m)$ be the subset of $A$ consisting of all elements that have property $P_{i}$. Then the total number of elements in $A$ that have exactly $r$ properties, denoted by $A(r)$, has the following formula $(r \geqslant 0)$:
$$\begin{array}{l} A(r)=\left(r_{1}\right) \sum_{1 \leqslant i_{1}<i_{2}<\cdots<i_{r} \leqslant m}\left|A_{i_{1}} \cap A_{i_{2}} \cap \cdots \cap A_{i_{r}}\right| \\ -\left({ }_{r}^{r+1}\right) \sum_{1 \leqslant j_{1}<j_{2}<\cdots<j_{r+1} \leqslant m}\left|A_{j_{1}} \cap A_{j_{2}} \cap \cdots \cap A_{j_{r+1}}\right| \\ +-\cdots+(-1)^{m-r}\binom{m}{r}\left|A_{1} \cap A_{2} \cap \cdots \cap A_{m}\right| . \end{array}$$