If \(n(\mathcal{P}(A))=32\) and \(n(U)=13\), how many 2-element subsets does \(A'\) have?
Answer and explanation
Correct answer: 28
For a finite set with \(n(A)\) elements, its power set has \(2^{n(A)}\) elements. Thus, \(2^{n(A)}=32=2^5\), so \(n(A)=5\). Since the universal set has 13 elements, the complement has \(n(A')=13-5=8\) elements. The number of 2-element subsets of an 8-element set is \(\binom{8}{2}=\frac{8\times7}{2}=28\). Therefore, option B is correct.
Frequently asked questions
What is the correct answer to this question?
28
Why is this the correct answer?
For a finite set with \(n(A)\) elements, its power set has \(2^{n(A)}\) elements. Thus, \(2^{n(A)}=32=2^5\), so \(n(A)=5\). Since the universal set has 13 elements, the complement has \(n(A')=13-5=8\) elements. The number of 2-element subsets of an 8-element set is \(\binom{8}{2}=\frac{8\times7}{2}=28\). Therefore, 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.
