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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
Given: Total number of students = 200
Number of students study Mathematics = 120
Number of students study Physics = 90
Number of students study Chemistry = 70
Number of students study Mathematics and Physics = 40
Number of students study Mathematics and Chemistry = 50
Number of students study Physics and Chemistry = 30
Number of students study none of them = 20
Let U be the total number of students, P, M and C be the number of students study Physics, Mathematics and Chemistry respectively
To find: Number of students who study all the three subjects n(M ∩ P ∩ C)
n(U) = 200
n(M) = 120
n(P) = 90
n(C) = 70
n(M ∩ P) = 40
n(M ∩ C) = 50
n(P ∩ C) = 30
Number of students who play either of them = 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(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 (i) and (ii)
160 + n(P ∩ M ∩ C) = 180
⇒ n(P ∩ M ∩ C) = 180 – 160
⇒ n(P ∩ M ∩ C) = 20
Hence, there are 20 students who study all three subjects.
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