In a survey, n(U)=210, n(A)=96, n(B)=88, n(C)=74, n(A∩B)=39, n(B∩C)=31, n(C∩A)=28, and n(A∩B∩C)=14. How many people are in none of the sets?
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Answer and explanation
Correct answer: 36
Apply inclusion–exclusion to find the number in at least one set: n(A∪B∪C)=96+88+74−39−31−28+14=174. Pairwise intersections are subtracted because they were counted twice, and the triple intersection is added once because it was over-subtracted. The number outside all three sets is 210−174=36. Hence option A is correct.
Frequently asked questions
What is the correct answer to this question?
36
Why is this the correct answer?
Apply inclusion–exclusion to find the number in at least one set: n(A∪B∪C)=96+88+74−39−31−28+14=174. Pairwise intersections are subtracted because they were counted twice, and the triple intersection is added once because it was over-subtracted. The number outside all three sets is 210−174=36. 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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