AMC8 1991

AMC8 1991 · Q23

AMC8 1991 · Q23. It mainly tests Inclusion–exclusion (basic).

The Pythagoras High School band has 100 female and 80 male members. The Pythagoras High School orchestra has 80 female and 100 male members. There are 60 females who are members in both band and orchestra. Altogether, there are 230 students who are in either band or orchestra or both. The number of males in the band who are NOT in the orchestra is

毕达哥拉斯高中乐队有100名女生和80名男生。毕达哥拉斯高中管弦乐队有80名女生和100名男生。有60名女生同时是乐队和管弦乐队的成员。总共有230名学生至少参加了一个乐队或管弦乐队。不在管弦乐队的乐队男生人数是

(A) 10 10

(B) 20 20

(D) 50 50

(E) 70 70

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): There area 100 females in the band, 80 in the orchestra, and 60 in both. Thus, there are $(100+80)-60=120$ females in at least one of the groups. Since the total is 230, then there are $230-120=110$ males in at least one of the groups. There are 80 males in band and 100 males in orchestra, thus to find the number of males in both, $(80+100)-?=110$. There are 70 in both. Finally, the number of males in band who are not in orchestra is $80-70=10$.

答案（A）：乐队中有 100 名女生，管弦乐队中有 80 名女生，两者都参加的有 60 名。因此，至少参加其中一个团体的女生人数为 $(100+80)-60=120$。总人数为 230，所以至少参加其中一个团体的男生人数为 $230-120=110$。乐队中有 80 名男生，管弦乐队中有 100 名男生，因此要找两者都参加的男生人数，有 $(80+100)-?=110$，可得两者都参加的为 70 人。最后，只在乐队而不在管弦乐队的男生人数为 $80-70=10$。

Topics

CP.004 Inclusion–exclusion (basic)