Combinatorial Identities in Negative Binomial mgf derivation

Question

I'm trying to understand how to derive negative binomial mgf and came accross this discussion

Deriving Moment Generating Function of the Negative Binomial?

I can't understand the step that assumes,

$$\binom {y+r-1}{r-1} = \binom{y+r-1}{y}$$

Thank you.

Answer 1

$$\binom{n}{k} = \frac{n!}{k! \, (n-k)!}.$$ This also implies $$\binom{n}{n-k} = \frac{n!}{(n-k)! \, k!}.$$ So $$\binom{n}{k} = \binom{n}{n-k}.$$ Now repeat this with $n = y+r-1$ and $k = y$.