Question

If set \( A \) contains 8 elements, then the number of subsets of \( A \) which contain at least 6 elements is:

• A. 28
• B. 73
• C. 37
• D. 82

Hint:

For counting subsets with a certain number of elements, use the binomial coefficient \( \binom{n}{k} \) where \( n \) is the total number of elements and \( k \) is the size of the subset.

Answer 1

The Correct Option is C

The number of subsets of a set with 8 elements is \( 2^8 = 256 \). We are asked to find the number of subsets that contain at least 6 elements. 

Step 1: Use the binomial coefficient to calculate the number of subsets with exactly 6, 7, and 8 elements: \( \binom{8}{6} + \binom{8}{7} + \binom{8}{8} = \frac{8 \times 7}{2 \times 1} + \frac{8}{1} + 1 = 28 + 8 + 1 = 37 \) 

Step 2: The total number of subsets with at least 6 elements is 37.