Question

Given the set $ S = \{ a, b, c, d, e, f \} $, find the total number of subsets with an odd number of elements.

• A. \( 16 \)
• B. \( 32 \)
• C. \( 64 \)
• D. \( 2^{6-1} = 32 \)

Hint:

For any set with \( n \) elements, the number of subsets with an odd number of elements is equal to half of the total number of subsets, \( \frac{2^n}{2} \).

Answer 1

The Correct Option is B

The set \( S = \{ a, b, c, d, e, f \} \) has 6 elements. The total number of subsets of a set of size \( n \) is \( 2^n \). 
For this set, the total number of subsets is: \[ 2^6 = 64 \] Now, we need to find the number of subsets with an odd number of elements. By symmetry, half of the subsets will have an odd number of elements, and half will have an even number of elements. 
Therefore, the number of subsets with an odd number of elements is: \[ \frac{64}{2} = 32 \] 
Thus, the correct answer is (B) \( 32 \).