Vandermonde's Identity: How to find a closed formula for the given summation I've stumbled upon the following challenging question. Find a closed formula for the following summation $$ S(a,b,n) = \sum_{k=0}^n {a \choose k} {b \choose n-k}$$ for all possible parameters $a,b$.
Anyone happens to see how to solve this one?

One way to solve this is by using generating series.  With generating series, we barely have to think about what we are doing, and arrive at the answer nicely. Consider $$\sum_{n=0}^{\infty}\sum_{k=0}^{n}\binom{a}{k}\binom{b}{n-k}x^{n}.$$
Switching the order of summation, this becomes
$$\sum_{k=0}^{\infty}\binom{a}{k}\sum_{n=k}^{\infty}\binom{b}{n-k}x^{n}=\sum_{k=0}^{\infty}\binom{a}{k}x^{k}\sum_{n=0}^{\infty}\binom{b}{n}x^{n}.$$  Then, by the binomial theorem this is $$\left(1+x\right)^{a}\left(1+x\right)^{b}=(1+x)^{a+b}.$$
Since the $n^{th}$ coefficient of this is $\binom{a+b}{n}$, we see that $$\sum_{k=0}^n\binom{a}{k}\binom{b}{n-k}=\binom{a+b}{n}.$$
Hope that helps,
