<s>[INST] Prove that $\sum^{n}_{r=0}(-1)^r\cdot \large\frac{\binom{n}{r}}{\binom{r+3}{r}} = \frac{3!}{2(n+3)}$ 
Prove that $\displaystyle \sum^{n}_{r=0}(-1)^r\cdot \large\frac{\binom{n}{r}}{\binom{r+3}{r}} = \frac{3!}{2(n+3)}$

$\bf{My\; Try::}$ We can write $$\frac{\binom{n}{r}}{\binom{r+3}{r}} = \frac{n!}{r!\times (n-r)!}\times \frac{r!\times 3!}{(r+3)!} = \frac{3!\times n!}{(n-r)!\times (r+3)!}$$
So our sum is $$ 3!\sum^{n}_{r=0}(-1)^r\frac{n!}{(n-r)!\times (r+3)!} = \frac{3!}{(n+1)(n+2)(n+3)}\sum^{n}_{r=0}(-1)^r\frac{(n+3)!}{(r+3)!\times (n-r)!}$$
So we get Sum $$ \frac{3!}{(n+1)(n+2)(n+3)}\sum^{n}_{r=0}(-1)^r\binom{n+3}{r+3} = -\frac{3!}{(n+1)(n+2)}\sum^{n}_{r=0}(-1)^{r+3}\binom{n+3}{r+3}$$
$$  = -\frac{3!}{(n+1)(n+2)(n+3)}\left[-\binom{n+3}{3}+\binom{n+3}{4}-\binom{n+3}{5}--.......+(-1)^{n+3}\binom{n+3}{n+3}\right]$$
$$=-\frac{3!}{(n+1)(n+2)(n+3)}\left[\binom{n+3}{0}-\binom{n+3}{1}+\binom{n+3}{2}-\binom{n+3}{3}+\binom{n+3}{4}-\binom{n+3}{5}--.......+(-1)^{n+3}\binom{n+3}{n+3}\right]+\frac{3!}{(n+1)(n+2)(n+3)}\left[\binom{n+3}{0}-\binom{n+3}{1}+\binom{n+3}{2}\right]$$
So we get $$ = \frac{3!}{(n+1)(n+2)(n+3)}\left[1-(n+3)+\frac{(n+3)(n+2)}{2}\right] = \frac{3!}{(n+1)(n+2)}\times \frac{n^2+3n+2}{2} = \frac{3!}{(n+1)(n+2)(n+3)}\times \frac{(n+1)(n+2)}{2} = \frac{3!}{2(n+3)}$$
Can we prove it Using Combinatorial way,bcz above method is very lengthy
If yes then plz explain here
Thanks 
 [/INST] Your computational argument can be written a bit more simply:
$$\begin{align*}
\sum_{r=0}^n(-1)^r\frac{\binom{n}r}{\binom{r+3}r}&=3!\sum_{r=0}^n(-1)^r\frac{n!}{(n-r)!(r+3)!}\\
&=\frac6{(n+3)(n+2)(n+1)}\sum_{r=0}^n(-1)^r\binom{n+3}{r+3}\tag{1}\\
&=\frac6{(n+3)(n+2)(n+1)}\sum_{r=3}^{n+3}(-1)^{r-3}\binom{n+3}r\\
&=\frac6{(n+3)(n+2)(n+1)}\sum_{r=0}^n\left(\sum_{r=0}^{n+3}(-1)^{r-3}\binom{n+3}r-\sum_{r=0}^2(-1)^{r-3}\binom{n+3}r\right)\\
&=\frac6{(n+3)(n+2)(n+1)}\sum_{r=0}^2(-1)^{r-2}\binom{n+3}r\\
&=\frac6{(n+3)(n+2)(n+1)}\left(\binom{n+3}0-\binom{n+3}1+\binom{n+3}2\right)\\
&=\frac6{(n+3)(n+2)(n+1)}\left(1-(n+3)+\frac12(n+3)(n+2)\right)\\
&=\frac{6\left(\frac12(n+3)-1\right)}{(n+3)(n+1)}\\
&=\frac{3n+9-6}{(n+1)(n+3)}\\
&=\frac3{n+3}\;.
\end{align*}$$
For a slightly neater computational argument we can start by taking advantage of $(1)$ to rewrite the identity as 
$$\sum_{r=0}^n(-1)^r\binom{n+3}{r+3}=\binom{n+3}3\cdot\frac3{n+3}=\binom{n+2}2\;.\tag{2}$$
Now
$$\begin{align*}
\sum_{r=0}^n(-1)^r\binom{n+3}{r+3}&=\sum_{r=0}^n(-1)^r\left(\binom{n+2}{r+2}+\binom{n+2}{r+3}\right)\\
&=\sum_{r=0}^n(-1)^r\binom{n+2}{r+2}+\sum_{r=0}^n(-1)^r\binom{n+2}{r+3}\\
&=\sum_{r=0}^n(-1)^r\binom{n+2}{r+2}+\sum_{r=1}^{n+1}(-1)^{r-1}\binom{n+2}{r+2}\\
&=\binom{n+2}2+(-1)^n\binom{n+2}{n+3}\\
&=\binom{n+2}2\;.
\end{align*}$$
It’s fairly easy to see what’s going on here combinatorially, though what follows is too informal to be a true combinatorial argument. The righthand side of $(2)$ of course is the number of $2$-element subsets of $[n+2]$. The lefthand side of $(2)$ is
$$\binom{n+3}3-\binom{n+3}4+\binom{n+3}5-+\ldots\;.$$
The first term counts the $3$-element subsets of $[n+3]$; those that contain $n+3$ are in one-to-one correspondence with the $2$-element subsets of $[n+2]$, so we’d like to throw away the $3$-element subsets of $[n=3]$ that don’t contain $[n+3]$. These are in one-to-one correspondence with the $4$-element subsets of $[n+3]$ that do contain $n+3$, so as a next approximation we’ll throw away all $4$-element subsets of $[n+3]$, getting $\binom{n+3}3-\binom{n+3}4$.
Now, of course, we need to add back in the $4$-element subsets of $[n+3]$ that don’t contain $n+3$. These are in one-to-one correspondence with the $5$-element subsets of $[n+3]$ that do contain $n+3$, so as a next approximation we add back in all $5$-element subsets of $[n+3]$, getting $\binom{n+3}3-\binom{n+3}4+\binom{n+3}5$.
At this point it’s clear that we have an informal induction argument going, and a bit of thought shows that it ends in the right place.
</s>