In three sets, n(A∪B∪C)=172. The numbers of elements only in A, only in B, only in C, only in A∩B, only in B∩C, and only in C∩A are 38, 34, 29, 21, 18, and 16, respectively. What is n(A∩B∩C)?
Answer and explanation
Correct answer: 16
A three-set Venn diagram has seven regions inside the union: three regions belonging to exactly one set, three regions belonging to exactly two sets, and the central region belonging to all three sets. The six given regions total 38 + 34 + 29 + 21 + 18 + 16 = 156. Since the union contains all seven regions, the central region is 172 − 156 = 16. Therefore, n(A∩B∩C)=16.
Frequently asked questions
What is the correct answer to this question?
16
Why is this the correct answer?
A three-set Venn diagram has seven regions inside the union: three regions belonging to exactly one set, three regions belonging to exactly two sets, and the central region belonging to all three sets. The six given regions total 38 + 34 + 29 + 21 + 18 + 16 = 156. Since the union contains all seven regions, the central region is 172 − 156 = 16. Therefore, n(A∩B∩C)=16.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
