If n(A) = 48, n(B) = 43, n(C) = 41, n(A ∪ B ∪ C) = 99, n(A ∩ B) = 17, n(B ∩ C) = 14, and n(C ∩ A) = 13, what is n(A ∩ B ∩ C)?
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Answer and explanation
Correct answer: 11
For three sets, the inclusion–exclusion formula is n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C). Substituting the values gives 99 = 48 + 43 + 41 − 17 − 14 − 13 + x = 88 + x. Therefore, x = 11, so the central intersection contains 11 elements.
Frequently asked questions
What is the correct answer to this question?
11
Why is this the correct answer?
For three sets, the inclusion–exclusion formula is n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C). Substituting the values gives 99 = 48 + 43 + 41 − 17 − 14 − 13 + x = 88 + x. Therefore, x = 11, so the central intersection contains 11 elements.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
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