2024 AMC 12A Problem 16
Problem 16 of 25IntermediateCounting & Probability
A set of tokens — red, white, blue, and black — is to be distributed at random to game players, tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as where and are relatively prime positive integers. What is
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Solution
Treat all tokens as distinct; the total number of ways to deal to each player is For the favorable event, choose which player gets the reds, whites, and blue in ways. The red player needs more token, the white player more, and the blue player more, all black; the black tokens split as in ways. So the probability is Then Thus, the correct answer is C.