Problem:
A set of 12 tokens- 3 red, 2 white, 1 blue, and 6 black-is to be distributed at random to 3 game players, 4 tokens per player. The probability that some player gets all the red tokens, another player 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 ?
Answer Choices:
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Solution:
The situation can be modeled by arranging the 12 tokens in a row, with the first player receiving the first 4 tokens, the second player receiving the next 4 tokens, and the third player receiving the last 4 tokens. There are such arrangements. The given conditions are satisfied if and only if the red tokens appear among the first 4 positions, the white tokens appear among the next 4 positions, and the blue token appears among the last 4 positions, or some permutation of the groups of 4. There are ! ways for this to happen. The required probability is therefore
and the requested sum is .
The problems on this page are the property of the MAA's American Mathematics Competitions