How to simplify this binomial coefficients?

Question

For $a, b, r, n \in \mathbb N$ : I have to simplify

$$\sum_{k = 0}^{r}\binom {a}k\binom {b}{r-k}$$

I tried by using factorials but it seemed more complicated

Answer 1

Hint:

Use that, by definition, $\dbinom nk$ is the coefficients of $x^k$ in the expansion of $(1+x)^n$. So here, consider the expansion of each side in the equality and compare the coefficients of $x^r$: $$(1+x)^a(1+x)^b=(1+x)^{a+b}.$$