The pigeonhole principle

#pigeonhole#Dirichlet#basic

Statement

Show that among $13$ people, at least two were born in the same month.

Hint

There are $12$ months; $13$ people.

Show solutionHide solution

Pigeonhole. Each person is a pigeon; each month is a hole. With $13$ pigeons and $12$ holes, some hole contains at least $2$ pigeons.

Conclusion. At least two people share a birth month.