If n(A ∪ B ∪ C) = 205, n(A ∩ B ∩ C) = 24, and exactly two sets contain a total of 83 elements, how many elements are in exactly one set?
Answer and explanation
Correct answer: 98
The union can be partitioned into three disjoint categories: elements in exactly one set, elements in exactly two sets, and elements in all three sets. Let the first category have x elements. Then 205 = x + 83 + 24. Therefore x = 205 − 83 − 24 = 98. The phrase “exactly two sets” means the total of the three pair-only regions, while the 24 elements in all three sets form a separate category. Hence option B is correct.
Frequently asked questions
What is the correct answer to this question?
98
Why is this the correct answer?
The union can be partitioned into three disjoint categories: elements in exactly one set, elements in exactly two sets, and elements in all three sets. Let the first category have x elements. Then 205 = x + 83 + 24. Therefore x = 205 − 83 − 24 = 98. The phrase “exactly two sets” means the total of the three pair-only regions, while the 24 elements in all three sets form a separate category. Hence option B is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
