Question

A set contains 9 elements. Then the number of subsets of the set which contains at most 4 elements is:

• A. 32
• B. 64
• C. 128
• D. 256
• E. 512

Hint:

For odd $n$, the sum of the first half of binomial coefficients is always $2^{n-1}$. Here $2^{9-1} = 2^8 = 256$.

Answer 1

The Correct Option is D

To determine the number of subsets of a set with 9 elements that contain at most 4 elements, we first need to understand a few key concepts and calculations regarding subsets.

A set with \( n \) elements has a total of \( 2^n \) subsets. For a set containing 9 elements, the total number of subsets is given by:

\(2^9 = 512\)

This total includes all subsets, ranging from the empty set (0 elements) to the full set (9 elements).

We need subsets containing at most 4 elements. These include sets with 0, 1, 2, 3, and 4 elements. We use the formula for combinations to calculate the number of subsets for each case:

• \(\binom{9}{0} = 1\): 1 subset with 0 elements (the empty set).
• \(\binom{9}{1} = 9\): 9 subsets with 1 element.
• \(\binom{9}{2} = 36\): 36 subsets with 2 elements.
• \(\binom{9}{3} = 84\): 84 subsets with 3 elements.
• \(\binom{9}{4} = 126\): 126 subsets with 4 elements.

Adding these numbers gives us the total number of subsets with at most 4 elements:

\(1 + 9 + 36 + 84 + 126 = 256\)

Therefore, the correct answer is 256.