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2025 AMC 12B 第 11 题

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所有题目均经美国数学协会（MAA）官方合法授权使用。

11.

九名运动员参加篮球队选拔，且没有两人身高相同。他们依次从一个袋子中随机抽取腕带，不放回；袋中有 33 条蓝色、33 条红色、33 条绿色腕带。他们被分成蓝组、红组和绿组。每组最高的成员被指定为该组队长。三名队长正好是最高的三名运动员的概率是多少？

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 33 blue bands, 33 red bands, and 33 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?

29\dfrac{2}{9}

27\dfrac{2}{7}

928\dfrac{9}{28}

13\dfrac{1}{3}

38\dfrac{3}{8}

答案：C
知识点：基本概率对称性
难度评级：1590
小提示：

队长是最高的三人，当且仅当这三人分到三个不同的组。

The captains are the three tallest exactly when the three tallest land in three different groups

大提示：

依次放置前三高的人；每个人都必须避开已经用过的组，而剩余位置数也在减少。

Place the tallest three one at a time; each must avoid the groups already used, whose slots shrink the pool

解答：

每组有 33 个位置。最高的三名运动员正好成为队长，当且仅当他们落在三个不同的组中。将三人依次放入 99 个位置，第二人在剩下的 88 个位置中有 66 个可选，第三人在剩下的 77 个位置中有 33 个可选。因此概率为 68⋅37=928\dfrac{6}{8} \cdot \dfrac{3}{7} = \dfrac{9}{28}。

所以正确答案是 C。

Each group has 33 slots. The three tallest athletes are the captains precisely when they fall into three different groups. Placing them one at a time into the 99 slots, the second must avoid the first’s group (66 of the remaining 88 slots) and the third must avoid both used groups (33 of the remaining 77 slots). The probability is 68⋅37=928.\dfrac{6}{8} \cdot \dfrac{3}{7} = \dfrac{9}{28}.

Thus, the correct answer is C.

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