Prove of an addition theorem for the general binomial coefficients

Question

Prove that:

$\sum_{k=0}^n \binom{s}{k} \binom{t}{n - k} = \binom{s + t}{n}$

for all $s, t \in\Bbb C $, $n \in N\cup {0}$.

That's pretty much all I'm given, and therefore, I haven't come quite far yet. I thought about first proving it for all $s, t \in \Bbb N$, and then somehow arguing that it must be the case for complex s, t too.

Because of n being a natural number, I've thought about proving it via induction; however, I'm not sure on how to do the induction step and how to rearrange it in order to be true for all n. If there's another way of proving it besides induction, I'm open to it aswell.

Thanks in advance for any help.

Answer 1

Here is one way. We have $(1+x)^{s+t} = \sum_{n=0}^\infty \binom{s+t}{n} x^n$. Similarly for $(1+x)^t$ and $(1+x)^s$.

Now use the fact that $(1+x)^{s+t} = (1+x)^s (1+x)^t$ and equate coefficients of $x^n$.