For three sets A, B, and C, n(A) = 72, n(B) = 66, and n(C) = 60. Exactly one set contains 84 elements, and all three sets together contain 10 elements. How many elements belong to exactly two sets?
Answer and explanation
Correct answer: 42
Let x be the number of elements belonging to exactly two sets. The sum of the three set cardinalities is 72 + 66 + 60 = 198. Elements in exactly one set contribute once, elements in exactly two sets contribute twice, and elements in all three sets contribute three times. Thus 198 = 84 + 2x + 3(10) = 114 + 2x, so 2x = 84 and x = 42. Therefore, option B is correct.
Frequently asked questions
What is the correct answer to this question?
42
Why is this the correct answer?
Let x be the number of elements belonging to exactly two sets. The sum of the three set cardinalities is 72 + 66 + 60 = 198. Elements in exactly one set contribute once, elements in exactly two sets contribute twice, and elements in all three sets contribute three times. Thus 198 = 84 + 2x + 3(10) = 114 + 2x, so 2x = 84 and x = 42. 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.
