Question

If $ \sum_{k=0}^{n+1} C_k^n = 512 $, find $ \sum_{k=0}^{n} C_k^n $.

• A. \( 256 \)
• B. \( 512 \)
• C. \( 1024 \)
• D. \( 1023 \)

Hint:

For binomial coefficients, remember that the sum \( \sum_{k=0}^{n} C_k^n \) always equals \( 2^n \), and consider the properties of binomial coefficients when solving such problems.

Answer 1

The Correct Option is A

The given summation is: \[ \sum_{k=0}^{n+1} C_k^n = 512 \] We know that the sum of the binomial coefficients for a fixed \( n \) is equal to \( 2^n \), which means: \[ \sum_{k=0}^{n} C_k^n = 2^n \] Now, we are given that: \[ \sum_{k=0}^{n+1} C_k^n = 512 \] This can be written as: \[ \sum_{k=0}^{n} C_k^n + C_{n+1}^n = 512 \] Since \( C_{n+1}^n = 1 \) (based on the property of binomial coefficients), we get: \[ 2^n + 1 = 512 \] Solving for \( n \), we get: \[ 2^n = 511 \] Therefore, \( n = 9 \). Thus: \[ \sum_{k=0}^{n} C_k^n = 2^9 = 512 \] Finally, the correct answer is (A) \( 256 \).