Question

In a survey where 100 students reported which subjects they like, 32 students in total liked Mathematics, 38 students liked Business and 30 students liked Literature. Moreover 7 students liked both Mathematics and Literature, 10 students liked both Mathematics and Business, 8 students liked both Business and Literature, 5 students liked all three subjects.

Then the number of people who liked exactly one subject is

1. 60
2. 65
3. 70
4. 78

Answer 1

Correct Answer - Option 2 : 65

Concept:

Number of elements present in exactly only one set

n(A) + n(B) + n(C) – 2 × [n(A ⋂ B) + n(B ⋂ C) + n(A ⋂ C)] + 3 × n(A ⋂ B ⋂ C)]

 

Calculations:

Given, U : Total number of students.

n(U) = 100

M  : Student who liked mathematics

n(M) = 32

B  : Student who liked Business

n(B) = 38

L  : Student who liked  Literature

n(B) = 30

Moreover,  students liked both Mathematics and Literature = n(\(\rm M\cap L\)) = 7

students liked both Mathematics and Business = n(\(\rm M\cap B\)) = 10

students liked both Business and Literature = n(\(\rm B\cap L\)) = 8

students liked all three subjects = n(\(\rm M\cap B\cap L\)) = 5

Now, number of people who liked exactly one subject 

= n(M) + n(B) + n(L) – 2 × [n(M ⋂ B) + n(B ⋂ L) + n(M ⋂ L)] + 3 × n(M ⋂ B ⋂ L)]

= 32 + 38 + 30 - 2 (10 + 8 + 7) + 3(5)

= 65

Hence, option (2) is correct.