Is this $\binom{n}{p}$ for $p>n$ make a sense in mathematics or it is $0$ by convention? It is well known that gamma function is not defined at negative integers , but my question is to know how i take the value of   $\binom{n}{p}$ for  $p>n$ then is this  make a sense or it is  $0$ by convention ? 

If you directly define $\binom{n}{p}$ as $\frac{\Gamma(n+1)}{\Gamma(p+1)\Gamma(n-p+1)}$ and recall that $\Gamma(x)$ ha simple poles at $0,-1,-2,\ldots$, it is no wonder that $\binom{n}{p}=0$ for $p,n\in\mathbb{N}$ and $p>n$. It is usually introduced by convention, but it is also the unique convention which agrees with the analytic continuation of the binomial coefficients through the $\Gamma$ function.
