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Question 8 : Out of 80 students who appeared for the school exams in Mathematics (M), Physics (P) and Chemistry (C), 50 passed M , 30 passed P and 40 passed C. At most 20 students passed M and P at most 20 students passed P and C and at most 20 students passed C and M. The maximum number of students who could have passed all three exams is __________.

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M ∪ P ∪ C = 80 (assume 80 to get max value of x)

M ∪ P ∪ C = M + P + C - (M ∩ P) - (P ∩ C) - (M ∩ C)+ M ∩ P ∩ C

80 = 50 + 30 + 40 - (M ∩ P) - (P ∩ C) - (M ∩ C) + x

(M ∩ P) + (P ∩ C) + (M ∩ C) = 40 + x

At most 20

So, 20 + 20 + 20 = 40 + x

x_{max} = 20

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