2025 AMC 12B Problem 11
Attempt Problem 11 of the 2025 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 12B solutions, or check the answer key.
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11.
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 blue bands, red bands, and 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?
Answer: C
Small Hint:
The captains are the three tallest exactly when the three tallest land in three different groups
Big Hint:
Place the tallest three one at a time; each must avoid the groups already used, whose slots shrink the pool
Solution:
Each group has slots. The three tallest athletes are the captains precisely when they fall into three different groups. Placing them one at a time into the slots, the second must avoid the first’s group ( of the remaining slots) and the third must avoid both used groups ( of the remaining slots). The probability is
Thus, the correct answer is C.
Problem 11 in Other Years
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