If n(A∪B∪C)=184, n(A)=86, n(B)=91, n(C)=88 and n(A∩B∩C)=23, what is n(A∩B)+n(B∩C)+n(C∩A)?
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Answer and explanation
Correct answer: 104
Let S=n(A∩B)+n(B∩C)+n(C∩A). Inclusion–exclusion for three sets states n(A∪B∪C)=n(A)+n(B)+n(C)−S+n(A∩B∩C). Substituting gives 184=86+91+88−S+23=288−S. Hence S=288−184=104. Therefore option A is correct; 81 would result from mishandling the triple-intersection term.
Frequently asked questions
What is the correct answer to this question?
104
Why is this the correct answer?
Let S=n(A∩B)+n(B∩C)+n(C∩A). Inclusion–exclusion for three sets states n(A∪B∪C)=n(A)+n(B)+n(C)−S+n(A∩B∩C). Substituting gives 184=86+91+88−S+23=288−S. Hence S=288−184=104. Therefore option A is correct; 81 would result from mishandling the triple-intersection term.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
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