In a survey, n(U) = 200, n(A) = 96, n(B) = 88, n(C) = 74, n(A ∩ B) = 40, n(B ∩ C) = 31, n(C ∩ A) = 29, and n(A ∩ B ∩ C) = 12. How many people are in none of the sets?
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Answer and explanation
Correct answer: 30
Apply inclusion–exclusion 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). Thus the union is 96 + 88 + 74 − 40 − 31 − 29 + 12 = 170. The universal set has 200 people, so those in none of the sets are 200 − 170 = 30. Hence option C is correct.
Frequently asked questions
What is the correct answer to this question?
30
Why is this the correct answer?
Apply inclusion–exclusion 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). Thus the union is 96 + 88 + 74 − 40 − 31 − 29 + 12 = 170. The universal set has 200 people, so those in none of the sets are 200 − 170 = 30. Hence option C 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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