2025 AMC 10B Problem 14

Problem 14 of 25IntermediateCounting & Probability

Nine athletes, no two of whom are the same height, try out for the basketball team. One at a time, they draw a wristband at random, without replacement, from a bag containing

3

blue bands,

3

red bands, and

3

green bands. They are divided into a blue group, a red group, and a green group. The tallest member of each group is named the group captain. What is the probability that the group captains are the three tallest athletes?

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Solution

The captains are the three tallest exactly when those three land in three different groups, since each is then the tallest of its own group. Drop them into the

9

slots one at a time (

3

per group). The second tallest misses the first’s group with probability

\tfrac{6}{8},

and the third misses both with probability

\tfrac{3}{7}.

So the probability is

\tfrac{6}{8} \cdot \tfrac{3}{7} = \tfrac{9}{28}.

Therefore, the answer is C.

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