# In a survey of 200 students of a school,

Question:

In a survey of 200 students of a school, it was found that 120 study Mathematics, 90 study Physics and 70 study Chemistry, 40 study Mathematics and Physics,

30 study Physics and Chemistry, 50 study Chemistry and Mathematics and 20 none of these subjects. Find the number of students who study all the three

subjects.

Solution:

According to the question,

Total number of students = n(U) = 200

Number of students who study Mathematics = n(M) = 120

Number of students who study Physics = n(P) = 90

Number of students who study Chemistry = n(C) = 70

Number of students who study Mathematics and Physics = n(M ∩ P) = 40

Number of students who study Mathematics and Chemistry = n(M ∩ C) = 50

Number of students who study Physics and Chemistry = n(P ∩ C) = 30

Number of students who study none of them = 20

Let the total number of students = U

Let the number of students who study Mathematics = M

Let the number of students who study Physics = P

Let the number of students who study Chemistry = C

number of students who study all the three subjects n(M ∩ P ∩ C)

Number of students who play either of them = n(P ∪ M ∪ C)

n(P ∪ M ∪ C) = Total – none of them

= 200 – 20

= 180 …(i)

Number of students who play either of them = n(P ∪ M ∪ C)

n(P ∪ M ∪ C) = n(C) + n(P) + n(M) – n(M ∩ P) – n(M ∩ C) – n(P ∩ C) + n(P ∩ M ∩ C)

= 120 + 90 + 70 – 40 – 30 – 50 + n(P ∩ M ∩ C)

= 160 + n(P ∩ M ∩ C) …(ii)

From equation (i) and (ii), we get,

160 + n(P ∩ M ∩ C) = 180

⇒ n(P ∩ M ∩ C) = 180 – 160

⇒ n(P ∩ M ∩ C) = 20

Therefore, there are 20 students who study all the three subjects.