2020 AMC 8 第 23 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。
23.
五个不同的奖项要颁给三名学生。每名学生至少获得一个奖项。共有多少种不同的颁奖方式？
Five different awards are to be given to three students. Each student will receive at least one award. In how many different ways can the awards be distributed?
小提示：
先从无条件的 种分配开始
Start with unrestricted distributions
大提示：
对没有学生获奖的情况使用容斥
Use inclusion-exclusion for students receiving no awards
视频讲解：
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文字解答：
若没有“每人至少一个奖项”的限制，每个奖项有 个学生可选， 个奖项共有 种分配。
减去至少一名学生没有获奖的情况。指定某名学生没有获奖时，五个奖项只能给另外两人，有 种；三名学生都可能是没有获奖者，所以先减去 。不过所有奖项都给同一个人的 种情况被多减了一次。
由容斥，符合条件的分配数为 。
正确答案是 B。
There are ways to give each of the distinct awards to one of the students.
Subtract the distributions in which at least one student receives no award. If a particular student receives none, the awards go to the other two students in ways. This gives counts, but the cases in which one student receives all awards have each been subtracted twice.
By inclusion-exclusion, the desired number is
Thus, the correct answer is B.
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