AMC 8 2020 Problem 23

AMC8 2020 年第 23 题

排列与组合★★★★☆

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?

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

正确答案 · Correct Answer

解析 · Solution

解法1

常规解法 · 按人数划分两类人数划分只有

\{3,1,1\}

与

\{2,2,1\}

两种。

\{3,1,1\}

：选得 3 份的人 3 种 × 选 3 份

\binom53=10

× 余 2 份各给两人 2!=2，得 60；

\{2,2,1\}

：选得 1 份的人 3 种 × 选 1 份

\binom51=5

× 余 4 份两人各 2

\binom42=6

，得 90。合计 150。

Solution 1

Elementary · split by award-count partition The award-count partitions are

\{3,1,1\}

and

\{2,2,1\}

. For

\{3,1,1\}

: 3 ways to pick who gets 3 awards,

\binom53=10

ways to choose them, and 2!=2 ways to give the remaining 2 awards one each — 60. For

\{2,2,1\}

: 3 ways to pick who gets 1 award,

\binom51=5

for it, and

\binom42=6

to split the last 4 two-and-two — 90. Total 60+90=150.

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