If n(A) = 82, n(B) = 76, n(C) = 70, n(A ∪ B ∪ C) = 162, n(A ∩ B) = 34, n(B ∩ C) = 30, and n(C ∩ A) = 27, what is n(A ∩ B ∩ C)?
Answer and explanation
Correct answer: 15
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 given values gives 162 = 82 + 76 + 70 − 34 − 30 − 27 + x = 147 + x. Therefore x = 15, so n(A ∩ B ∩ C) is 15. The triple intersection is added because it was subtracted three times in the pairwise terms.
Frequently asked questions
What is the correct answer to this question?
15
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 given values gives 162 = 82 + 76 + 70 − 34 − 30 − 27 + x = 147 + x. Therefore x = 15, so n(A ∩ B ∩ C) is 15. The triple intersection is added because it was subtracted three 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.
