AMC10 2017 B

AMC10 2017 B · Q13

AMC10 2017 B · Q13. It mainly tests Inclusion–exclusion (basic).

There are 20 students participating in an after-school program offering classes in yoga, bridge, and painting. Each student must take at least one of these three classes, but may take two or all three. There are 10 students taking yoga, 13 taking bridge, and 9 taking painting. There are 9 students taking at least two classes. How many students are taking all three classes?

有20名学生参加课后项目，提供瑜伽、桥牌和绘画课程。每名学生至少选一门课，但可选两门或三门。有10名学生选瑜伽，13名选桥牌，9名选绘画。有9名学生至少选两门课。选三门课的学生有多少名？

(A) 1 1

(B) 2 2

(D) 4 4

(E) 5 5

Answer

Correct choice: (C)

正确答案：（C）

Solution

Let x, y, and z be the number of people taking exactly one, two, and three classes, respectively. The condition that each student in the program takes at least one class is equivalent to the equation x + y + z = 20. The condition that there are 9 students taking at least two classes is equivalent to the equation y + z = 9. The sum 10 + 13 + 9 = 32 counts once the students taking one class, twice the students taking two classes, and three times the students taking three classes. Then x + 2y + 3z = 32, which is equivalent to z = 32 − (x + y + z) − (y + z) = 32 − 20 − 9 = 3.

设x、y、z分别为选恰好一门、两门、三门课的人数。每个学生至少选一门课，等价于x + y + z = 20。有9名至少选两门，等价于y + z = 9。总和10 + 13 + 9 = 32计算了一次选一门的学生，两次选两门，三次选三门。然后x + 2y + 3z = 32，等价于z = 32 − (x + y + z) − (y + z) = 32 − 20 − 9 = 3。

Topics

CP.004 Inclusion–exclusion (basic)