Given n(U) = 180, n(A) = 76, n(B) = 69, n(C) = 62, n(A ∩ B) = 30, n(B ∩ C) = 27, n(C ∩ A) = 24, and n(A ∩ B ∩ C) = 10, how many elements belong to none of the sets?
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Answer and explanation
Correct answer: 44
Use the inclusion-exclusion formula for three sets: 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 76 + 69 + 62 − 30 − 27 − 24 + 10 = 136. Elements in none of the sets are 180 − 136 = 44. Hence, option A is correct.
Frequently asked questions
What is the correct answer to this question?
44
Why is this the correct answer?
Use the inclusion-exclusion formula for three sets: 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 76 + 69 + 62 − 30 − 27 − 24 + 10 = 136. Elements in none of the sets are 180 − 136 = 44. Hence, option A is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
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