The general inclusion-exclusion formula

Formula.
$$\left|\bigcup_{i=1}^{n}A_i\right|=\sum_{\emptyset\neq I\subseteq\{1,\ldots,n\}}(-1)^{|I|+1}\left|\bigcap_{i\in I}A_i\right|.$$

Expanded. $\sum_i|A_i|-\sum_{i<j}|A_i\cap A_j|+\sum_{i<j<k}|A_i\cap A_j\cap A_k|-\cdots+(-1)^{n+1}|A_1\cap\cdots\cap A_n|$.

Conclusion. The formula alternates signs according to the size of the intersection index set.