2025 AMC 12B Problem 11

Problem 11 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

Each group has

3

slots. The three tallest athletes are the captains precisely when they fall into three different groups. Placing them one at a time into the

9

slots, the second must avoid the first’s group (

6

of the remaining

8

slots) and the third must avoid both used groups (

3

of the remaining

7

slots). The probability is

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

Thus, the correct answer is C.

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