Question

Let \(A = \{a,b,c,d,e,f\}\) Then the number of subsets of \(A\) with an odd number of elements is

• A. 8
• B. 12
• C. 16
• D. 24
• E. 32

Hint:

For any non-empty set, number of subsets of even size = number of odd size = \(2^{n-1}\).

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
Total subsets = \(2^6 = 64\). Half have odd cardinality, half even. So odd = 32.

Step 2: Detailed Explanation:
By symmetry, \(\sum_{k \text{ odd}} \binom{6}{k} = 2^{5} = 32\).

Step 3: Final Answer:
Option (E).