<s>[INST] Sum of binomial coefficients involving $n,p,q,r$ 
Sum of binomial product $\displaystyle \sum^{n}_{r=0}\binom{p}{r}\binom{q}{r}\binom{n+r}{p+q}$

Simplifying $\displaystyle \frac{p!}{r!\cdot (p-r)!} \cdot \frac{q!}{r!\cdot (q-r)!}\cdot \frac{(n+r)!}{(p+q)! \cdot (n+r-p-q)!}$.
Could some help me with this, thanks
 [/INST] The proposed identity can be derived (putting $r=s$) from this more general one
$$
\sum\limits_{\left( {0\, \le } \right)\,k\,\left( { \le \,n} \right)} {\left( \matrix{
  m - \left( {r - s} \right) \cr 
  k \cr}  \right)\left( \matrix{
  n + \left( {r - s} \right) \cr 
  n - k \cr}  \right)\left( \matrix{
  r + k \cr 
  m + n \cr}  \right)}  = \left( \matrix{
  r \cr 
  m \cr}  \right)\left( \matrix{
  s \cr 
  n \cr}  \right)\quad \quad \left| \matrix{
  \,{\rm 0} \le {\rm integer \, }m,n \hfill \cr 
  \;{\rm real}\;r,s \hfill \cr}  \right.
$$
reported in "Concrete Mathematics" - pag. 171, and which is demonstrated through the following passages:
$$
\begin{gathered}
  \sum\limits_{\left( {0\, \leqslant } \right)\,k\,\left( { \leqslant \,n} \right)} {\left( \begin{gathered}
  m - \left( {r - s} \right) \\ 
  k \\ 
\end{gathered}  \right)\left( \begin{gathered}
  n + \left( {r - s} \right) \\ 
  n - k \\ 
\end{gathered}  \right)\left( \begin{gathered}
  r + k \\ 
  m + n \\ 
\end{gathered}  \right)}  =  \hfill \\
   = \sum\limits_{\begin{subarray}{l} 
  \left( {0\, \leqslant } \right)\,k\,\left( { \leqslant \,n} \right) \\ 
  \left( {0\, \leqslant } \right)\,j\,\left( { \leqslant \,n} \right) 
\end{subarray}}  {\left( \begin{gathered}
  m - \left( {r - s} \right) \\ 
  k \\ 
\end{gathered}  \right)\left( \begin{gathered}
  n + \left( {r - s} \right) \\ 
  n - k \\ 
\end{gathered}  \right)\left( \begin{gathered}
  r \\ 
  m + n - j \\ 
\end{gathered}  \right)\left( \begin{gathered}
  k \\ 
  j \\ 
\end{gathered}  \right)}  =  \hfill \\
   = \sum\limits_{\begin{subarray}{l} 
  \left( {0\, \leqslant } \right)\,k\,\left( { \leqslant \,n} \right) \\ 
  \left( {0\, \leqslant } \right)\,j\,\left( { \leqslant \,m + n} \right) 
\end{subarray}}  {\left( \begin{gathered}
  m - \left( {r - s} \right) \\ 
  k \\ 
\end{gathered}  \right)\left( \begin{gathered}
  k \\ 
  j \\ 
\end{gathered}  \right)\left( \begin{gathered}
  n + \left( {r - s} \right) \\ 
  n - k \\ 
\end{gathered}  \right)\left( \begin{gathered}
  r \\ 
  m + n - j \\ 
\end{gathered}  \right)}  =  \hfill \\
   = \sum\limits_{\begin{subarray}{l} 
  \left( {0\, \leqslant } \right)\,k\,\left( { \leqslant \,n} \right) \\ 
  \left( {0\, \leqslant } \right)\,j\,\left( { \leqslant \,m + n} \right) 
\end{subarray}}  {\left( \begin{gathered}
  m - \left( {r - s} \right) \\ 
  j \\ 
\end{gathered}  \right)\left( \begin{gathered}
  m - \left( {r - s} \right) - j \\ 
  k - j \\ 
\end{gathered}  \right)\left( \begin{gathered}
  n + \left( {r - s} \right) \\ 
  n - k \\ 
\end{gathered}  \right)\left( \begin{gathered}
  r \\ 
  m + n - j \\ 
\end{gathered}  \right)}  =  \hfill \\
   = \sum\limits_{\left( {0\, \leqslant } \right)\,j\,\left( { \leqslant \,m + n} \right)} {\left( \begin{gathered}
  m - \left( {r - s} \right) \\ 
  j \\ 
\end{gathered}  \right)\left( \begin{gathered}
  m + n - j \\ 
  n - j \\ 
\end{gathered}  \right)\left( \begin{gathered}
  r \\ 
  m + n - j \\ 
\end{gathered}  \right)}  =  \hfill \\
   = \sum\limits_{\left( {0\, \leqslant } \right)\,j\,\left( { \leqslant \,m + n} \right)} {\left( \begin{gathered}
  m - \left( {r - s} \right) \\ 
  j \\ 
\end{gathered}  \right)\left( \begin{gathered}
  m + n - j \\ 
  m \\ 
\end{gathered}  \right)\left( \begin{gathered}
  r \\ 
  m + n - j \\ 
\end{gathered}  \right)}  =  \hfill \\
   = \sum\limits_{\left( {0\, \leqslant } \right)\,j\,\left( { \leqslant \,m + n} \right)} {\left( \begin{gathered}
  m - \left( {r - s} \right) \\ 
  j \\ 
\end{gathered}  \right)\left( \begin{gathered}
  r \\ 
  m \\ 
\end{gathered}  \right)\left( \begin{gathered}
  r - m \\ 
  n - j \\ 
\end{gathered}  \right)}  =  \hfill \\
   = \left( \begin{gathered}
  r \\ 
  m \\ 
\end{gathered}  \right)\left( \begin{gathered}
  s \\ 
  n \\ 
\end{gathered}  \right) \hfill \\ 
\end{gathered}
$$
consisting in:  


*

*Vandermonde de-convolution

*shift of the 4th binomial

*"trinomial revision"

*sum on $k$ by Vandermonde convolution

*symmetry

*"trinomial revision"

*sum on $j$ by Vandermonde convolution

</s>