This

principle was first stated by

Dirichlet in 1834 and has

important applications in

number theory.

*If m pigeons are put into m pigeonholes, there is an empty hole if and only if there's a hole with more than one pigeon.*
It is also known as the

Dirichlet's Box Principle, which reads:

*If n>m pigeons are put into m pigeonholes, there's a hole with more than one pigeon. *
Here's a simple

application: if we have 37 students in our

classroom, then at least 4

students must have their birthdays in the same month. (The pigeonholes are the months, so n = 12; clearly 37 = 12ยท3 + 1 so r = 3.)

A similar

principle to the

Pigeonhole Principle is the

Fubini Principle:

*If the average number of envelopes per pigeonhole is a, then some pigeonhole will have at least a envelopes. Similarly, there must be a pigeonhole with at most a envelopes.*