In a survey, n(U) = 260, n(A) = 118, n(B) = 104, n(C) = 92, n(A ∩ B) = 48, n(B ∩ C) = 41, n(C ∩ A) = 37, and n(A ∩ B ∩ C) = 16. How many are in none of the sets?
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Answer and explanation
Correct answer: 56
Apply 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 118 + 104 + 92 − 48 − 41 − 37 + 16 = 204. Those in none of the sets are 260 − 204 = 56, so option B is correct.
Frequently asked questions
What is the correct answer to this question?
56
Why is this the correct answer?
Apply 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 118 + 104 + 92 − 48 − 41 − 37 + 16 = 204. Those in none of the sets are 260 − 204 = 56, so option B 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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