For three sets A, B, and C, n(A) = 84, n(B) = 78, and n(C) = 72. There are 96 elements in exactly one set and 12 elements in all three sets. How many elements are in exactly two sets?
Answer and explanation
Correct answer: 51
Add the cardinalities of the three sets to obtain the total membership count: 84 + 78 + 72 = 234. If x elements belong to exactly two sets, then elements in exactly one set contribute 96, elements in exactly two sets contribute 2x, and elements in all three sets contribute 3 × 12. Hence 234 = 96 + 2x + 36, so 2x = 102 and x = 51. Therefore, option B is correct.
Frequently asked questions
What is the correct answer to this question?
51
Why is this the correct answer?
Add the cardinalities of the three sets to obtain the total membership count: 84 + 78 + 72 = 234. If x elements belong to exactly two sets, then elements in exactly one set contribute 96, elements in exactly two sets contribute 2x, and elements in all three sets contribute 3 × 12. Hence 234 = 96 + 2x + 36, so 2x = 102 and x = 51. Therefore, option B is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
