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Problem 657

AMC 10/12, early questions

Number theory Difficulty 3.0 Prove it CEMC Euclid · Canada · 2002

Three bins are labelled A, B and C, and each bin contains four balls numbered 1, 2, 3, and 4. The balls in each bin are mixed, and then a student chooses one ball at random from each of the bins. If $a$ , $b$ and $c$ are the numbers on the balls chosen from bins A, B and C, respectively, the student wins a toy helicopter when $a = b + c$ . There are $64$ ways to choose the three balls. What is the probability that the student wins the prize?

This one wants a proof. Work it on paper, then check yourself against the publisher's own solution, linked below. Be honest about it: the record is only any use to you if it is.

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Source: CEMC, University of Waterloo, licensed CC-BY-NC-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project. Solutions are the publisher's, linked not copied.