If n(A)=40, n(B)=36, n(C)=34, n(A∪B∪C)=82, n(A∩B)=12, n(B∩C)=10, and n(C∩A)=9, what is n(A∩B∩C)?
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Answer and explanation
Correct answer: 3
For three sets, 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=n(A∩B∩C). Substitution gives 82=40+36+34−12−10−9+x, or 82=79+x. Hence x=3. The common central region of all three sets therefore contains 3 elements, so option A is correct.
Frequently asked questions
What is the correct answer to this question?
3
Why is this the correct answer?
For three sets, 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=n(A∩B∩C). Substitution gives 82=40+36+34−12−10−9+x, or 82=79+x. Hence x=3. The common central region of all three sets therefore contains 3 elements, so 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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