**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.