JEE Main 2024 Mathematics question
A group of 40 students appeared in an examination of 3 subjects – Mathematics, Physics and Chemistry. It was found that all students passed in atleast one of the subjects, 20 students passed in Mathematics, 25 students passed in Physics, 16 students passed in Chemistry, atmost 11 students passed in both Mathematics and Physics, atmost 15 students passed in both Physics and Chemistry, atmost 15 students passed in both Mathematics and Chemistry. The maximum number of students passed in all the three subjects is .
Solution
Let a = n(M∩P) ≤ 11, b = n(P∩C) ≤ 15, c = n(M∩C) ≤ 15 and x = n(M∩P∩C). From n(M∪P∪C) = 40 = 61 − (a + b + c) + x we get x = a + b + c − 21. The regions 'only in P∩C', 'only in M∩C' and 'only in M∩P' must be non-negative: b − x ≥ 0, c −
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