What is the count of odd-sized subsets for a set of size 5?

The correct answer is Option 4 (16).

Key Points

• A set of size 5 has 2⁵ = 32 subsets in total.
• Out of these 32 subsets, half of them (16 subsets) will have an odd size, and the other half will have an even size. This is because for any set, the total subsets are evenly distributed between odd-sized subsets and even-sized subsets.
• Odd-sized subsets are subsets with sizes 1, 3, or 5 (since these are the odd numbers ≤ 5).
• The counts of subsets for each odd size can be calculated using the binomial coefficient formula C(n, k) = n! / [k!(n-k)!], where n is the size of the set, and k is the size of the subset:
  • Subsets of size 1: C(5, 1) = 5
  • Subsets of size 3: C(5, 3) = 10
  • Subsets of size 5: C(5, 5) = 1
• Adding these counts together: 5 + 10 + 1 = 16.

Additional Information

• The binomial coefficient C(n, k) represents the number of ways to choose k items from a set of n items.
• The formula for calculating subsets ensures that all combinations are accounted for, including both odd and even sizes.
• For sets of even size, the number of odd-sized subsets always equals the number of even-sized subsets.
• This property is a consequence of the symmetry of the binomial coefficients in Pascal's Triangle.
• For sets of odd size, there will always be one more odd-sized subset than even-sized subsets.