If n(A ∪ B ∪ C) = 162, exactly one set contains 73 elements and exactly two sets contain 56 elements, then what is n(A ∩ B ∩ C)?
Answer and explanation
Correct answer: 33
The union A ∪ B ∪ C is partitioned into three mutually exclusive categories: elements in exactly one set, elements in exactly two sets, and elements in all three sets. Hence n(A ∪ B ∪ C) = exactly-one + exactly-two + all-three. Substituting the data gives 162 = 73 + 56 + n(A ∩ B ∩ C). Therefore n(A ∩ B ∩ C) = 162 − 73 − 56 = 33.
Frequently asked questions
What is the correct answer to this question?
33
Why is this the correct answer?
The union A ∪ B ∪ C is partitioned into three mutually exclusive categories: elements in exactly one set, elements in exactly two sets, and elements in all three sets. Hence n(A ∪ B ∪ C) = exactly-one + exactly-two + all-three. Substituting the data gives 162 = 73 + 56 + n(A ∩ B ∩ C). Therefore n(A ∩ B ∩ C) = 162 − 73 − 56 = 33.
Which subject and chapter does this question cover?
This is a Class 12 Mathematics question. Chapter: Sets.
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