I think many of those reading this will be familiar with the idea of a combinatorial argument: instead of proving some combinatorial identity (usually involving binomial coefficents) by algebraic manipulation, one can construct an analogy (bijection?) in terms of sets, committees etc. and use that to prove the identity.
I'm curious to know some techniques that can easily be used to prove stuff using combinatorial arguments. A big-list for this would consist of some answers, each sketching out one such strategy and providing an example (somewhat like this question on PPCG). Would it be an on-topic question?
