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 __________.
Explanation:
To maximise the number of students who passed in all the three exams we will assume that no one failed in all the three exams.
∴ M ∪ P ∪ C = M + P + C - M ∩ P - P ∩ C - C ∩ M + M ∩ P ∩ C
⇒ M ∪ P ∪ C - (M + P + C) + M ∩ P + P ∩ C + C ∩ M = M ∩ P ∩ C
To maximise the number of students who passed in all the three exams we need to maximise M ∩ P and P ∩ C and C ∩ M. Maximum value of M ∩ P =20, that of P ∩ C =20 and that of C ∩ M = 20
∴ 80 - (50 + 30 + 40) + 20 + 20 + 20 + M ∩ P ∩ C
⇒ 20 = M ∩ P ∩ C
Hence, 20.
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