Theorem (Pigeonhole principle) Let be a positive integer. If more than objects are distribuited among containers, then some container must contain more than one object. Proof Suppose by contradiciont that in all containers there is at most one object, then in total we would have at most objects.
We can say more if is a lot bigger than :
Theorem (Generalized pigeonhole principle) If we distribuite more than object in containers, then there exists at least one which contains objects. Proof As before, suppose by contraddiction that each container at most has objects. Then in total we have less or equal objects.