Exercise 532Combinatorics › Pigeonhole principleHigh schoolAdded 3 Nov 2026

Balls in boxes

#generalized pigeonhole#ceil#balls

Statement

How many balls must be placed in $7$ boxes to guarantee that some box contains at least $4$ balls?

Hint

If every box has $\le 3$ balls, at most $7\cdot 3=21$ balls fit.

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Maximum without guarantee. Each box has $\le3$ : total $\le21$ .

Guarantee. With $22$ balls, by generalized pigeonhole some box has at least $\lceil 22/7\rceil=4$ balls.

Conclusion. $22$ balls.