Question

Of the students in a certain class, 10 had taken a course in A, 11 had taken a course in

B, and 14 had taken a course in C. If 3 students had taken a course in all of the A, B,

and C, and 20 students had taken a course in only one of A, B, and C, how many

students had taken a course in exactly two of A, B, and C?​

Answer 1

Given : Students in a certain class, 10 had taken a course in A, 11 had taken a course in  B, and 14 had taken a course in C. If 3 students had taken a course in all of the A, B,  and C, and 20 students had taken a course in only one of A, B, and C,

To Find : how many  students had taken a course in exactly two of A, B, and C?​

Solution:

Students only in A =  n(A) - n(A∩B) - n(A∩C)  + n(A∩B∩C) = 10 - (n(A∩B) +n(A∩C) ) + 3  = 13 - (n(A∩B) +n(A∩C) )

Students only in B =  n(B) - n(A∩B) - n(A∩C)  + n(A∩B∩C) = 11 - (n(A∩B) +n(B∩C) ) + 3  = 14 - (n(A∩B) +n(B∩C) )

Students only in C =  n(C) - n(A∩B) - n(A∩C)  + n(A∩B∩C) = 14 - (n(A∩C) +n(B∩C) ) + 3  = 17 - (n(A∩C) +n(B∩C) )

13 - (n(A∩B) +n(A∩C) )  +  14 - (n(A∩B) +n(B∩C) )  + 17 - (n(A∩C) +n(B∩C) )

= 20

=> 2((n(A∩B) + n(A∩C) + n(B∩C) ) = 24

=> n(A∩B) + n(A∩C) + n(B∩C)  = 12

students had taken a course in exactly two of A, B, and C

=  n(A∩B)  - n(A∩B∩C)  + n(A∩C) - n(A∩B∩C)  + n(B∩C)  - n(A∩B∩C)

=  n(A∩B) + n(A∩C) + n(B∩C)  -  3n(A∩B∩C)

= 12 - 3(3)

= 3

3 students had taken a course in exactly two of A, B, and C

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