AMC12 2016 A

AMC12 2016 A · Q11

AMC12 2016 A · Q11. It mainly tests Probability (basic), Casework.

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?

某夏令营有100名学生，每个学生会唱歌、跳舞或表演中的至少一种。有些学生有不止一种才艺，但没有学生同时具备三种才艺。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。有多少名学生具备其中两种才艺？

(A) 16 16

(B) 25 25

(D) 49 49

(E) 64 64

Answer

Correct choice: (E)

正确答案：（E）

Solution

Answer (E): Because 42 students cannot sing, $100-42=58$ can sing. Similarly, $100-65=35$ can dance, and $100-29=71$ can act. This gives a total of $58+35+71=164$. However, the students with two talents have been counted twice in this sum. Because there are 100 students in all, $164-100=64$ students must have been counted twice.

答案（E）：因为有42名学生不会唱歌，所以会唱歌的有$100-42=58$人。同理，会跳舞的有$100-65=35$人，会表演的有$100-29=71$人。总计为$58+35+71=164$。然而，有两种才能的学生在这个总和中被计算了两次。因为总共有100名学生，所以$164-100=64$名学生一定被重复计算了。

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