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

Step 1: Understanding the Concept:
The number of subsets with exactly $k$ elements from a set of $n$ elements is given by the binomial coefficient $\binom{n}{k}$.
Step 2: Key Formula or Approach:
Required number $= \binom{n}{0} + \binom{n}{1} + \dots + \binom{n}{4}$ for $n=9$.
Step 3: Detailed Explanation:
Total subsets for $n=9$ is $2^9 = 512$.
Sum of all coefficients: $\sum_{k=0}^{9} \binom{9}{k} = 512$.
By symmetry: $\binom{9}{0} = \binom{9}{9}$, $\binom{9}{1} = \binom{9}{8}$, $\binom{9}{2} = \binom{9}{7}$, $\binom{9}{3} = \binom{9}{6}$, $\binom{9}{4} = \binom{9}{5}$.
Thus, $\binom{9}{0} + \dots + \binom{9}{4}$ is exactly half of the total sum.
Result $= 512 / 2 = 256$.
Step 4: Final Answer:
The number of subsets is 256.