Question

If the value of the sum $29\left(\genfrac{}{}{0ex}{}{30}{0}\right)+28\left(\genfrac{}{}{0ex}{}{30}{1}\right)+27\left(\genfrac{}{}{0ex}{}{30}{2}\right)+.....+1\left(\genfrac{}{}{0ex}{}{30}{28}\right)+0\left(\genfrac{}{}{0ex}{}{30}{29}\right)-\left(\genfrac{}{}{0ex}{}{30}{30}\right)$ where $\left(\begin{array}{c}n\\ r\end{array}\right)={}^n{C}_r$, is equal to $k{2}^{32}$, then the value of $k$ is equal to

• A. 7
• B. 14
• C. $\frac{5}{2}$
• D. $\frac{7}{2}$

Answer 1

Answer: (D)

$S=29\left(\genfrac{}{}{0ex}{}{30}{0}\right)+28\left(\genfrac{}{}{0ex}{}{30}{1}\right)+27\left(\genfrac{}{}{0ex}{}{30}{2}\right)+.....+1\left(\genfrac{}{}{0ex}{}{30}{28}\right)+0\left(\genfrac{}{}{0ex}{}{30}{29}\right)-\left(\genfrac{}{}{0ex}{}{30}{30}\right)$
Also $S=-\left(\genfrac{}{}{0ex}{}{30}{0}\right)+0\left(\genfrac{}{}{0ex}{}{30}{1}\right)+1\left(\genfrac{}{}{0ex}{}{30}{2}\right)+.......+27\left(\genfrac{}{}{0ex}{}{30}{28}\right)+28\left(\genfrac{}{}{0ex}{}{30}{29}\right)+29\left(\genfrac{}{}{0ex}{}{30}{30}\right)$
$\Rightarrow 2S=28\left({}^{30}{C}_0+{}^{30}{C}_1+{}^{30}{C}_2+\dots \dots \dots +{}^{30}{C}_{30}\right)=28\left(2^{30}\right)$
Hence, $S=14\times 2^{30}=\frac{7}{2}\times 2^{32}.\Rightarrow k=\frac{7}{2}$