<s>[INST] Prove that this sum of products of binomial coefficient can be simplified like this $a , b, c$ are positive integers such that ${a}\geq{b+c}$.
Prove that $$\sum_{i=0}^{b}{\binom{a}{i+c}\binom{b}{i}}=\binom{a+b}{b+c}$$
 [/INST] $\binom{a}{i+c}$ is the coefficient of $x^{i+c}$ from $(x+1)^{a}$ while $\binom{b}{i}$ is the coefficient of $\frac{1}{x^{i}}$ from $(\frac{1}{x}+1)^{b}$.
$\sum_{i=0}^{b}\binom{a}{i+c}\binom{b}{i}$ is the coefficient of $x^{c}$ from $(x+1)^{a}(\frac{1}{x}+1)^b=\frac{(x+1)^{a+b}}{x^{b}}$ which is $\binom{a+b}{b+c}$.
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