把5个不同的奖章分配给3个学生，每个学生得到至少一个奖章。那么一共有多少种不同的分配方法？

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.