If n(A) = 46, n(B) = 40, n(C) = 37, n(A ∩ B) = 14, n(B ∩ C) = 12, n(C ∩ A) = 11, and n(A ∪ B ∪ C) = 91, what is n(A ∩ B ∩ C)?
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Answer and explanation
Correct answer: 5
Use the three-set inclusion–exclusion formula: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C). Substitution gives 91 = 46 + 40 + 37 − 14 − 12 − 11 + x = 86 + x. Hence x = 5. The central intersection is added back because it was subtracted too many times in the pairwise terms.
Frequently asked questions
What is the correct answer to this question?
5
Why is this the correct answer?
Use the three-set inclusion–exclusion formula: n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(B ∩ C) − n(C ∩ A) + n(A ∩ B ∩ C). Substitution gives 91 = 46 + 40 + 37 − 14 − 12 − 11 + x = 86 + x. Hence x = 5. The central intersection is added back because it was subtracted too many times in the pairwise terms.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
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