How to prove the binomial theorem

Question

I'm trying to prove the following:

$$ (p + q)^n = \sum_{x=1}^n \frac{n!}{x!(n-x)!}p^xq^{n-x}. $$

But I'm not sure where to start, would expanding the left hand side get me anywhere?

Any tips or hints to prove this ?

Answer 1

As Rodrigo Ribeiro said, you could try induction. Note that the following result will be useful:

$\binom{n}{k} + \binom{n}{k-1}$ = $\binom{n+1}{k}$

which can be proven algebraically.