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.

[![enter image description here][3]][3]


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