If n(A) = 5, how many subsets in P(A) have an odd number of elements?
Answer and explanation
Correct answer: 16
For every non-empty finite set with n elements, the numbers of even-cardinality and odd-cardinality subsets are equal. Since A has five elements, P(A) has 2⁵ = 32 subsets in total. Therefore the number with an odd number of elements is half of 32, or 2⁴ = 16. Equivalently, 5C1 + 5C3 + 5C5 = 5 + 10 + 1 = 16. Thus option C is correct.
Frequently asked questions
What is the correct answer to this question?
16
Why is this the correct answer?
For every non-empty finite set with n elements, the numbers of even-cardinality and odd-cardinality subsets are equal. Since A has five elements, P(A) has 2⁵ = 32 subsets in total. Therefore the number with an odd number of elements is half of 32, or 2⁴ = 16. Equivalently, 5C1 + 5C3 + 5C5 = 5 + 10 + 1 = 16. Thus option C is correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Power Set and Subsets.
