If \(A=\{1,2,3,4,5,6\}\), how many subsets have exactly two odd elements?
Answer and explanation
Correct answer: 24
The odd elements of \(A\) are \(1,3,5\), so there are three odd elements. To obtain a subset with exactly two odd elements, choose two of these three odd elements: \(\binom{3}{2}=3\) choices. Each of the three even elements, \(2,4,6\), may be independently included or excluded, giving \(2^3=8\) choices. By the multiplication principle, the total is \(\binom{3}{2}2^3=3\times8=24\).
Frequently asked questions
What is the correct answer to this question?
24
Why is this the correct answer?
The odd elements of \(A\) are \(1,3,5\), so there are three odd elements. To obtain a subset with exactly two odd elements, choose two of these three odd elements: \(\binom{3}{2}=3\) choices. Each of the three even elements, \(2,4,6\), may be independently included or excluded, giving \(2^3=8\) choices. By the multiplication principle, the total is \(\binom{3}{2}2^3=3\times8=24\).
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
