summing this binomial series

Question

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.

Answer 1

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, $$ \sum_{k=0}^7 \binom{7}{k} 2^{14-k} = 2^7\sum_{k=0}^{7}\binom{7}{k} 2^{7-k} = 2^7(1+2)^7 = 6^7\ . $$

Answer 2

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 $7$.

Answer 3

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.