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.