Combinatorial principles

Bijective proofs are utilized to demonstrate that two sets have the same number of elements.

The pigeonhole principle often ascertains the existence of something or is used to determine the minimum or maximum number of something in a discrete context.

Generating functions and recurrence relations are powerful tools that can be used to manipulate sequences, and can describe if not resolve many combinatorial situations.

Using this one can, for example, demonstrate the existence of some element in a set with some specific properties.

Recurrence relations may lead to previously unknown properties of a sequence, but generally closed-form expressions for the terms of a sequence are more desired.