In a survey, n(U)=180, n(A)=82, n(B)=76, n(C)=69, n(A∩B)=34, n(B∩C)=29, n(C∩A)=27, and n(A∩B∩C)=12. How many people are in none of the sets?
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Answer and explanation
Correct answer: 31
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). Thus, the union is 82+76+69−34−29−27+12=149. The people in none of the sets are outside the union, so the required number is n(U)−n(A∪B∪C)=180−149=31. Therefore, option A is correct.
Frequently asked questions
What is the correct answer to this question?
31
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). Thus, the union is 82+76+69−34−29−27+12=149. The people in none of the sets are outside the union, so the required number is n(U)−n(A∪B∪C)=180−149=31. Therefore, 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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