Exercise 446Combinatorics › Counting principlesHigh schoolAdded 3 Nov 2026

Pigeonhole with balls

#pigeonhole#balls#boxes

Statement

How many balls must be placed in $5$ boxes to guarantee at least one box contains at least $3$ balls?

Hint

Maximize the number of balls if every box has $\le2$ .

Show solutionHide solution

Max without guarantee. Each box has $\le2$ : at most $2\cdot5=10$ balls.

Guarantee. With $11$ balls, by pigeonhole some box has $\ge3$ .

Conclusion. $11$ balls.