Question

某夏令营的 $100$ 名学生每人都会唱歌、跳舞或表演。有些学生有不止一种才艺，但没有学生三种才艺都会。有 $42$ 名学生不会唱歌， $65$ 名学生不会跳舞， $29$ 名学生不会表演。有多少名学生正好有两种才艺？

Each of the $100$ students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are $42$ students who cannot sing, $65$ students who cannot dance, and $29$ students who cannot act. How many students have two of these talents?

$16$

$25$

$36$

$49$

$64$

答案：E

知识点：补集计数容斥原理双重计数

难度评级：1470

解答：

会唱歌、跳舞、表演的人数分别为 $100-42=58$ 、 $100-65=35$ 、 $100-29=71$ ，总和为 $58+35+71=164$ 。

因为没有人三种才艺都会，每个学生有一种或两种才艺，所以只有一种才艺的学生被数一次，有两种才艺的学生被数两次。因此，被重复计算的人数为 $164-100=64$ 。

所以正确答案是 E。

The numbers who can sing, dance, and act are $100-42=58,$ $100-65=35,$ and $100-29=71,$ respectively, for a total of $58+35+71=164.$

Since no student has all three talents, each student has one or two talents, so single-talent students are counted once and two-talent students are counted twice. The number counted twice is $164-100=64.$

Thus, the correct answer is E.

2016 AMC 12A 第 11 题

答案：E

难度评级：1470

解答：

其他年份的第 11 题