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2025 AMC 12A Problem 6

Problem 6 of 25EasierCombinatoricsProbability & Statistics

Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in. What is the probability that the two students will sit in two adjacent chairs and the two teachers will also sit in two adjacent chairs?

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Solution

Choosing 22 chairs for the students and 22 for the teachers gives (62)(42)=15⋅6=90\binom{6}{2}\binom{4}{2} = 15 \cdot 6 = 90 equally likely outcomes. A round table has 66 adjacent pairs of chairs. Give the students any adjacent pair; among the remaining 44 chairs there are exactly 33 adjacent pairs for the teachers. That is 6⋅3=186 \cdot 3 = 18 favorable outcomes. The probability is 1890=15.\dfrac{18}{90} = \dfrac{1}{5}. Thus, the correct answer is B.
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Tagged: circular arrangements · basic probability

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