Question

The value of \( 2^{15}C_1 + 2^{15}C_2 - 3^{15}C_3 + \cdots - 15^{15}C_{15} + \binom{14}{1} + \binom{14}{3} + \binom{14}{5} + \cdots + \binom{14}{11} \) is :

• A. $2^{16} - 1$
• B. $2^{13} - 14$
• C. $2^{13} - 13$
• D. $2^{14}$

Hint:

Remember the identity: $\sum_{r=0}^{n} ^nC_r = 2^n$ and the sum of alternate coefficients is $2^{n-1}$.

Answer 1

The Correct Option is B

Step 1: The sum $\sum_{r=1}^{n} (-1)^{r-1} \cdot r \cdot ^nC_r$ is related to the derivative of $(1-x)^n$ at $x=1$, which is 0 for $n>1$.
Step 2: Adjusting the indices and specific terms for $n=15$ and the additional series of $14C_r$.
Step 3: The sum of odd binomial coefficients $\sum ^{14}C_{2k+1} = 2^{14-1} = 2^{13}$.
Step 4: Subtracting the missing terms (like $14C_{13}$) and simplifying leads to $2^{13} - 14$.