If n(A)=70, n(B)=65, n(C)=60, n(A∪B∪C)=140, n(A∩B)=28, n(B∩C)=24, and n(C∩A)=22, what is n(A∩B∩C)?
Answer and explanation
Correct answer: 19
Let x=n(A∩B∩C). The inclusion–exclusion formula gives 140=70+65+60−28−24−22+x. The sum of the single-set cardinalities is 195, and the sum of the pairwise intersections is 74, so 140=121+x. Therefore x=19. The triple intersection must be added because subtracting all pairwise intersections removes the common central region too many times.
Frequently asked questions
What is the correct answer to this question?
19
Why is this the correct answer?
Let x=n(A∩B∩C). The inclusion–exclusion formula gives 140=70+65+60−28−24−22+x. The sum of the single-set cardinalities is 195, and the sum of the pairwise intersections is 74, so 140=121+x. Therefore x=19. The triple intersection must be added because subtracting all pairwise intersections removes the common central region too many times.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
