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?

A. 120

B. 150

C. 180

D. 210

E. 240

1. 2020 AMC 8 官方解析

There are two cases to consider.

Case 1 : One student receives 1 award and the other two students receive 2 awards each .
There are 3 ways to select the student who will receive 1 award and 5 choices for that student's award. That leaves 4 awards to distribute evenly. For the next student there are $C_4^2 = \binom{4}{2} = \frac{4 \cdot 3}{2} = 6$ ways to choose that student's 2 awards, leaving the remaining 2 awards for the third student. Therefore the number of ways to give 1 award to one student and 2 awards to each of the other two students is $3 \cdot 5 \cdot 6 = 90$.

Case 2 :
One student receives 3 awards and the other two students receive 1 award each.
There are 3 ways to select the student who receives 3 awards and $C_5^3 = \binom{5}{3} = \frac{5 \cdot 4 \cdot 3}{3 \cdot 2 \cdot 1} = 10$ ways to choose that student's awards. The remaining 2 awards can be distributed to the other two students in 2 ways. Therefore the number of ways to give 3 awards to one student and 1 award to each of the other two students is $3 \cdot 10 \cdot 2 = 60$. In total there are $90 + 60 = 150$ ways to distribute the 5 awards.

解答：有两种情况要考虑.

情况1：一名学生获得1个奖项，其他两名学生分别获得2个奖项. 有3种方式可以选择获得1个奖项的学生，有5种方式可以选择其获得的奖项. 然后剩下4个奖项平均分配给其他两名学生. 对于下一个学生，有$C_4^2 = \binom{4}{2} = \frac{4 \cdot 3}{2} = 6$ 种方式选择该学生获得的2个奖项，剩下的2个奖项则留给了第三位学生. 一名学生获得1个奖项，其他两名学生分别获得2个奖项的情况有 $3 \cdot 5 \cdot 6 = 90$ 中选择方式.

情况2：一名学生获得3个奖项，其他两名学生分别获得1个奖项. 有3种方式可以选择获得3个奖项的学生，有$C_5^3 = \binom{5}{3} = \frac{5 \cdot 4 \cdot 3}{3 \cdot 2 \cdot 1} = 10$ 种方法可以选择该学生的3个奖项. 其余的2个奖项可以有2种不同的方式分配给其他两个学生. 因此，一名学生获得3个奖项，其他两名学生分别获得1个奖项的情况有 $3 \cdot 10 \cdot 2 = 60$ 种选择方式. 因此总共有 $90 + 60 = 150$ 种方式分配5个奖项.