Question

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?

$120$

$150$

$180$

$210$

$240$

答案：B

知识点：容斥原理乘法原理

难度评级：1370

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 $3$ 个学生可选， $5$ 个奖项共有 $3^5=243$ 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 $2^5$ 种；三名学生都可能是没有获奖者，所以先减去 $3\cdot2^5$ 。不过所有奖项都给同一个人的 $3$ 种情况被多减了一次。

由容斥，符合条件的分配数为 $3^5-3\cdot2^5+3=150$ 。

正确答案是 B。

There are $3^5=243$ ways to give each of the $5$ distinct awards to one of the $3$ 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 $2^5$ ways. This gives $3\cdot2^5$ counts, but the $3$ cases in which one student receives all awards have each been subtracted twice.

By inclusion-exclusion, the desired number is $3^5-3\cdot2^5+3=150.$

Thus, the correct answer is B.

2020 AMC 8 第 23 题

答案：B

难度评级：1370

视频讲解：

文字解答：

其他年份的第 23 题