Question

在某所大学，数学科学学部由数学系、统计系和计算机科学系组成。每个系都有两名男教授和两名女教授。现在要组成一个由六名教授构成的委员会，其中必须有三名男性和三名女性，并且三个系中每个系都必须有两名教授入选。求满足这些要求的委员会有多少种。

At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible committees that can be formed subject to these requirements.

答案

答案：88

知识点：组合有限制的排列分类讨论

难度评级：2070

解答：

每个系恰好贡献两名委员会成员。如果每个系都派出一名男教授和一名女教授，则每个系有 $2 \cdot 2 = 4$ 种选择，共有 $4^3 = 64$ 个委员会。

否则，某个系派出两名男教授。为了保持男女各三名，另一个系必须派出两名女教授，剩下的系派出一男一女。选择全男性系有 $3$ 种方法，选择全女性系有 $2$ 种方法，混合系中有 $2 \cdot 2 = 4$ 种选择（全男性系和全女性系的人选都是唯一的），所以共有 $3 \cdot 2 \cdot 4 = 24$ 个委员会。

总数为 $64 + 24 = 88$ 。

Each department contributes exactly two committee members. If every department sends one man and one woman, there are $2 \cdot 2 = 4$ choices per department, for $4^3 = 64$ committees.

Otherwise some department sends two men. To keep three of each gender, another department must then send its two women, and the remaining department sends one man and one woman. There are $3$ ways to pick the all-male department, $2$ ways to pick the all-female department, and $2 \cdot 2 = 4$ choices in the mixed department (the two-man and two-woman selections are forced), for $3 \cdot 2 \cdot 4 = 24$ committees.

The total is $64 + 24 = 88.$

2012 AIME II 第 3 题

答案：88

难度评级：2070

解答：

其他年份的第 3 题