I'm trying to prove by induction the following:
$$\displaystyle \binom{a}{0}\binom{b}{n}+\binom{a}{1}\binom{b}{n-1}+...+\binom{a}{n}\binom{b}{0}=\binom{a+b}{n}$$.
As I didn't know how to prove by induction on two variables, I'm following this other question (which I think it was very helpful), however, I'm stuck on the second step:
$$\forall x,y. P(x+1,y) \Rightarrow P(x,y+1)$$ which in this case, it means I have to prove:
$$\displaystyle \binom{a}{0}\binom{b+1}{n}+\binom{a}{1}\binom{b+1}{n-1}+...+\binom{a}{n}\binom{b+1}{0}=\binom{a+b+1}{n}$$
(by the way, I switched the variables order so I'm using $P(b,a)$
Maybe ironically, I tried to prove that by induction (this time on a single variable), but I don't know to proceed about it.
Any help?
Regards