Summary of inclusion-exclusion

Two sets. $|A\cup B|=|A|+|B|-|A\cap B|$.

Three sets. $|A\cup B\cup C|=\sum|A_i|-\sum|A_i\cap A_j|+|A\cap B\cap C|$.

General.
$$\left|\bigcup_i A_i\right|=\sum_{\emptyset\neq I}(-1)^{|I|+1}\left|\bigcap_{i\in I}A_i\right|.$$

Derangements. $D_n=n!\sum_{k=0}^n(-1)^k/k!$.

Surjections. $\text{Surj}(m,n)=\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^m$.

Exactly $k$ properties. $E_k=\sum_{j=k}^n(-1)^{j-k}\binom{j}{k}N_j$.

Conclusion. IE is one of the most versatile counting tools, applicable to unions, complements, derangements, surjections, and beyond.