Proving $\sum_{m=0}^n\binom{n}{m}^2 \binom{m}{n-k}=\binom{n}{k}\binom{n+k}{k}$

Question

How can I prove this? $$\sum_{m=0}^n\binom{n}{m}^2 \binom{m}{n-k}=\binom{n}{k}\binom{n+k}{k}$$

$$ 0\le k \le n $$ I developed the expressions, but they are not the same. I do not know if it will be my mistake or the advisor I tried to solve for the binomial, but I could not, any idea to be able to proceed

Answer 1

You can use the general identity $$\sum_{k\ge0}\binom ak\binom bk\binom kc=\binom ac\binom{a+b-c}a$$ Inserting $n$ for $a$ and $b$ and $n-k$ for $c$ yields what you wanted to prove.
The upper bound of summation can be reduced from $\infty$ to $n$ because $\binom nk$ is 0 for $k\gt n$.

EDIT: This can be proven using Vandermonde's identity. @user gives a very simple proof in their answer.

Answer 2

$$\sum_{m=0}^{n}\binom{n}{m}^{2}\binom{m}{n-k}=\binom{n}{k}\sum_{m=0}^{n}\binom{n}{m}\binom{k}{n-m}=\binom{n}{k}\binom{n+k}{n}$$where the second equality rests on the vandermonde identity.

Answer 3

Let us prove the general form of the identity given by @Richard.

$$\sum_{r=0}^\infty \binom ar \binom br \binom rc = \binom ac \binom{a+b-c}a.$$

Replacing $\binom ar \binom rc$ with: $$ \binom ar \binom rc=\frac{a!}{r!(a-r)!}\frac{r!}{c!(r-c)!} =\frac{a!}{c!(a-c)!}\frac{(a-c)!}{(a-r)!(r-c)!}=\binom ac\binom {a-c}{a-r}, $$ one obtains: $$\sum_{r=0}^\infty \binom ar \binom br \binom rc = \binom ac\sum_{r=0}^\infty \binom br \binom {a-c}{a-r} =\binom ac \binom{a+b-c}a, $$ where in the last equality the Vandermonde's identity was used.

Inserting $a=b=n$ and $c=n-k$ yields the identity of OP, as already was pointed out by @Richard.