Exercise 259Combinatorics › Pigeonhole principleHigh schoolAdded 3 Nov 2026

The pigeonhole principle on birthdays

#pigeonhole#birthday#basic application

Statement

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

Hint

There are $12$ months; count pigeons (people) and holes (months).

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Setup. $13$ pigeons (people) and $12$ holes (months).

Pigeonhole. Since $13>12$ , by the pigeonhole principle some hole contains at least $2$ pigeons.

Conclusion. At least two people share a birth month.