Binomial coefficient expansion question

Question

I'm trying to follow this expansion (linked below) for one of my classes, but nothing I have tried is proving successful.

Any hints or help would be very appreciated.

Thanks :)

Binomial coefficient expansion

Answer 1

There is a mistake, here is the correct expansion $${n\choose r}=\frac{n!}{r!(n-r)!}=\frac{n}{n-r}\cdot\frac{(n-1)!}{r!(n-1-r)!}=\frac{n}{n-r}{n-1\choose r}$$