If n(A)=82, n(B)=77, n(C)=69, n(A∪B∪C)=151, n(A∩B)=32, n(B∩C)=28, and n(C∩A)=25, what is n(A∩B∩C)?
Answer and explanation
Correct answer: 8
The governing concept is inclusion–exclusion for three sets. For sets A, B, and C, n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C). Let x represent the triple intersection. Substituting the data gives 151=82+77+69−32−28−25+x. The calculation on the right before x is 228−85=143, so 151=143+x and x=8. Therefore option A is correct. The triple intersection must be added once because its elements were included in all three single-set counts and then removed through the three pairwise intersections. The other choices fail the stated formula.
Frequently asked questions
What is the correct answer to this question?
8
Why is this the correct answer?
The governing concept is inclusion–exclusion for three sets. For sets A, B, and C, n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C). Let x represent the triple intersection. Substituting the data gives 151=82+77+69−32−28−25+x. The calculation on the right before x is 228−85=143, so 151=143+x and x=8. Therefore option A is correct. The triple intersection must be added once because its elements were included in all three single-set counts and then removed through the three pairwise intersections. The other choices fail the stated formula.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
