If n(A)=25, n(B)=22, n(C)=19, n(A∩B)=7, n(B∩C)=6, n(C∩A)=5, n(A∩B∩C)=2, and n(U)=60, how many elements are in none of the sets?
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Answer and explanation
Correct answer: 10
For three sets, use the 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). Thus, n(A∪B∪C)=25+22+19−7−6−5+2=50. The elements in none of the sets are outside the union, so n(U)−n(A∪B∪C)=60−50=10. Therefore, option A is correct.
Frequently asked questions
What is the correct answer to this question?
10
Why is this the correct answer?
For three sets, use the 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). Thus, n(A∪B∪C)=25+22+19−7−6−5+2=50. The elements in none of the sets are outside the union, so n(U)−n(A∪B∪C)=60−50=10. 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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