Question

If \(\sum_{k = 0}^{n + 1} \binom{n+1}{k} = 512\) , then \(\sum_{k = 0}^{n} \binom{n}{k} =\)

• A. 512
• B. 256
• C. 511
• D. 510
• E. 128

Hint:

\(\sum_{k=0}^m \binom{m}{k} = 2^m\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
\(\sum_{k=0}^{n+1} \binom{n+1}{k} = 2^{n+1} = 512 = 2^9 \implies n+1 = 9 \implies n=8\). Then \(\sum_{k=0}^{8} \binom{8}{k} = 2^8 = 256\).

Step 2: Detailed Explanation:
Straightforward.

Step 3: Final Answer:
Option (B).