If U = {1, 2, ..., 180}, A is the set of multiples of 12, B is the set of multiples of 15, and C is the set of multiples of 20, what is n((A ∪ B ∪ C)′)?
Answer and explanation
Correct answer: 150
Use inclusion–exclusion. There are floor(180/12) = 15 multiples of 12, floor(180/15) = 12 multiples of 15, and floor(180/20) = 9 multiples of 20. The pairwise common counts are 3, 3, and 3, while the common triple count is 3. Thus the union has 15 + 12 + 9 − 3 − 3 − 3 + 3 = 30 elements. Its complement has 180 − 30 = 150 elements.
Frequently asked questions
What is the correct answer to this question?
150
Why is this the correct answer?
Use inclusion–exclusion. There are floor(180/12) = 15 multiples of 12, floor(180/15) = 12 multiples of 15, and floor(180/20) = 9 multiples of 20. The pairwise common counts are 3, 3, and 3, while the common triple count is 3. Thus the union has 15 + 12 + 9 − 3 − 3 − 3 + 3 = 30 elements. Its complement has 180 − 30 = 150 elements.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
