Show, by giving a proof by contradiction, that if $100$ balls are placed in nine boxes, some box contains 12 or more balls.
I would like to ask for a hint for this quesiton. Thank you.
2026-03-29 07:22:24.1774768944
Proof by Contradiction: $100$ Balls & $9$ Boxes
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The statement of the problem contains a very big hint: by giving a proof by contradiction. That means assuming that the result is false and trying to derive a contradiction from that assumption. What would make the result false? There would have to be some way to distribute $100$ balls amongst nine boxes so that no box contained $12$ or more balls, i.e., so that every box contained at most $11$ balls. What can you then conclude about the total number of balls in the nine boxes?