Question

If \( (1+x)^n = C_0 + C_1 x + C_2 x^2 + \dots + C_n x^n \), then \( C_0 + \frac{C_1}{2} + \frac{C_2}{3} + \dots + \frac{C_{10}}{11} \) is

• A. \( 2^{11} \)
• B. \( 2^{11} - 1 \)
• C. \( \frac{2^{11}}{11} \)
• D. None of these

Hint:

In binomial expansions, the sum of the coefficients at different powers can be computed using the symmetry of the expansion.

Answer 1

The Correct Option is B

Step 1: Understand the binomial expansion.
The binomial expansion of \( (1 + x)^n \) is given by \( C_0 + C_1 x + C_2 x^2 + \dots + C_n x^n \), where \( C_k \) are the binomial coefficients. The sum \( C_0 + \frac{C_1}{2} + \frac{C_2}{3} + \dots + \frac{C_{10}}{11} \) can be interpreted as a specific sum of terms from this expansion.

Step 2: Calculate the sum.
Using the properties of the binomial expansion, the value of the given sum is \( 2^{11} - 1 \). This result comes from evaluating the series formed by the coefficients and their corresponding factors.

Step 3: Conclusion.
Thus, the correct answer is \( 2^{11} - 1 \), which corresponds to option (B).