<s>[INST] Proof strategy to show $\large \sum_{0 \ \le \ x \ \le \ a} \normalsize \binom{a}{x} \binom{b}{n+x} = \binom{a+b}{a+n} $ The following two combinatorial identities are taken from a textbook. 
$$\begin{align}
&\large \sum_{0\ \le \ x \ \le \ n} \normalsize \binom{a}{x} \binom{b}{n-x} = \binom{a+b}{n} \tag{10} \\
&\large \sum_{0 \ \le \ x \ \le \ a} \normalsize \binom{a}{x} \binom{b}{n+x} = \binom{a+b}{a+n}\tag{11} \\
\end{align}$$ 
What proof strategy would you suggest to prove identity $(11)$, algebraically?  We may assume Vandermonde's Identity $(10)$; it may be useful in a proof of identity $(11)$.  
There appear to be many similarities between the two identities. How might we make use of that?
 [/INST] The first one: 
$$(1+x)^a (1+x)^b = (1+x)^{a+b}.$$
What is the coefficient of $x^n$ on the LHS? RHS?
Second one $$(1+1/x)^a(1+x)^b = (1+x)^a (1+x)^b/x^a.$$
What is the coefficient of $x^n$ on the LHS? RHS?
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