If \(A=\{1,2,3,4,5\}\), how many subsets of the power set \(\mathcal{P}(A)\) have even cardinality?
Answer and explanation
Correct answer: 16
The wording asks for subsets of \(A\) having even cardinality; these are the elements of the power set \(\mathcal{P}(A)\). Since \(|A|=5\), the number of even-cardinality subsets is \(\binom50+\binom52+\binom54=1+10+5=16\). Equivalently, for every nonempty set, even- and odd-cardinality subsets occur equally often, so each group has \(2^{5-1}=16\) subsets. Thus option B is correct.
Frequently asked questions
What is the correct answer to this question?
16
Why is this the correct answer?
The wording asks for subsets of \(A\) having even cardinality; these are the elements of the power set \(\mathcal{P}(A)\). Since \(|A|=5\), the number of even-cardinality subsets is \(\binom50+\binom52+\binom54=1+10+5=16\). Equivalently, for every nonempty set, even- and odd-cardinality subsets occur equally often, so each group has \(2^{5-1}=16\) subsets. Thus option B is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
