No doubts, the [inclusion-exclusion principle][1] generates most common type of formulae used in enumerative combinatorics. [Examples][2] include explicit formulae for derangements, Striling numbers, rook polynomials, Euler's totient function, and so on.

$$\left\lvert \bigcup_{i=1}^n A_i \right\rvert = \sum_{\emptyset\ne J\subseteq\{1,2,\dots,n\}} (-1)^{|J|-1} \left\lvert \bigcap_{j\in J}^n A_j \right\rvert$$

  [1]: https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle
  [2]: https://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle#Applications
  [3]: https://i.sstatic.net/1JWeR.png