summing this binomial series I found a really interesting question which is as follows:
Prove that the value of
$$\sum^{7}_{k=0}[({7\choose k}/{14\choose k})*\sum^{14}_{r=k}{r\choose k}{14\choose r}] = 6^7$$
my approach:
I tried to simplify the innermost sigma as well as  trying to simplify by using
${n\choose k}=n!/k!(n-k)!$ however I am can't get hold of this one.
My guess is that the summation simplifies into a standard series but I can't say for sure.
Kindly help me out.
 A: Using $${14 \choose r}{r \choose k} = {14 \choose k}{14-k \choose r-k}$$ given reduces to
$$
\begin{align*}
  & \sum_{k=0}^7 {7 \choose k} \bigg\{\sum^{14}_{r=k} {14-k \choose r-k} \bigg\}  \\
  & = \sum_{k=0}^7 {7 \choose k} \{2^{14-k}\} \\
  & = 2^{7} \times \sum_{k=0}^7 {7 \choose k} 2^{7-k} \\
  & = 2^{7}\times(2+1)^{7} \\
  & = 6^7 
\end{align*}
$$
Edit : As pointed by @ElliotYu, outer bound should be from $0$ to $7$.
A: First off I don't think your sum is quite right. The bounds on the outer sum should be $k=0$ to $7$, I believe, otherwise the value isn't $6^7$. (Question now corrected)
You are on the right track that rewriting the binomial coefficients in terms of factorials will help. Though the factors inside the sum over $r$ won't simplify much by themselves. The solution is to bring the factor $1/\binom{14}{k}$ into the second sum. This gives us
$$
\left.\frac{r!}{k!(r-k)!}\frac{14!}{r!(14-r)!}\right/\frac{14!}{k!(14-k)!} = \frac{(14-k)!}{(r-k)!(14-k)!}\ .
$$
This can be recognized as $\binom{14-k}{r-k}$.
Note that the inner sum is from $r = k$ to $14$, we can let $t = r-k$, and change the bounds to $0$ and $14-k$. This turns the inner sum into
$$
\sum_{t=0}^{14-k} \binom{14-k}{t} = 2^{14-k}\ .
$$
The outer sum can now be evaluated,
$$
\sum_{k=0}^7 \binom{7}{k} 2^{14-k} = 2^7\sum_{k=0}^{7}\binom{7}{k} 2^{7-k} = 2^7(1+2)^7 = 6^7\ .
$$
A: 
Setting $n=7$ we obtain
\begin{align*}
\color{blue}{\sum_{k=0}^n}&\color{blue}{\binom{n}{k}\binom{2n}{k}^{-1}\sum_{r=k}^{2n}\binom{r}{k}\binom{2n}{r}}\\
&=\sum_{k=0}^n\binom{n}{k}\frac{k!(2n-k)!}{(2n)!}\sum_{r=k}^{2n}\frac{r!}{k!(r-k)!}\,\frac{(2n)!}{r!(2n-r)!}\\
&=\sum_{k=0}^n\binom{n}{k}\sum_{r=k}^{2n}\binom{2n-k}{r-k}\\
&=\sum_{k=0}^n\binom{n }{k}\sum_{r=0}^{2n-k}\binom{2n-k}{r}\\
&=\sum_{k=0}^n\binom{n}{k}2^{2n-k}\\
&=2^{2n}\sum_{k=0}^n\binom{n}{k}\frac{1}{2^k}\\
&=2^{2n}\left(1+\frac{1}{2}\right)^n\\
&\,\,\color{blue}{=6^n}
\end{align*}
and the claim follows.

