Question

The value of $\sum _{k=0}^7[\frac{\left(\begin{array}{c}7\\ k\end{array}\right)}{\left(\begin{array}{c}14\\ k\end{array}\right)}\cdot \sum _{r=k}^{14}\left(\begin{array}{c}r\\ k\end{array}\right)\left(\begin{array}{c}14\\ r\end{array}\right)],$ where $\left(\begin{array}{c}n\\ r\end{array}\right)$ denotes ${}^n{C}_r$ is $a^b.$ Then the value of $a+b$ is (where $a$ and $b$ are coprime numbers)

Answer 1

$\sum _{k=0}^7[\frac{{}^7{C}_k}{{}^{14}{C}_k}\cdot \sum _{r=k}^{14}{}^r{C}_k\times {}^{14}{C}_r]$

$=\sum _{k=0}^7[\frac{7!\times k!\times (14-k)!}{14!\times k!\times (7-k)!}\cdot \sum _{r=k}^{14}\frac{r!}{k!(r-k)!}\times \frac{14!}{r!(14-r)!}]$

$=\sum _{k=0}^7({}^7{C}_k\cdot \sum _{r=k}^{14}{}^{14-k}{C}_{r-k})$
$=\sum _{k=0}^7({}^7{C}_k\cdot 2^{14-k})$
$=2^{14}\sum _{k=0}^7{}^7{C}_k(\frac{1}{2}{)}^k=2^{14}{(1+\frac{1}{2})}^7=6^7$
$\therefore a+b=13$