Reducing $\binom{t-2}{n-2}/\binom{t-1}{n-1}$ to $\frac{n-1}{t-1}$

Question

How does $$\frac{\displaystyle\binom{t-2}{n-2}}{\displaystyle\binom{t-1}{n-1}}$$ reduce to $$\frac{n-1}{t-1}$$

I know that the formula for the nCk = $$\frac{n!}{k!(n-k)!}$$

When i unfold given the formula i get

$$\frac{(t-2)!(n-1)!((t-1)-(n-1))!}{(n-2)!((t-2)-(n-2))!(t-1)!}$$ i dont see how this reduces and i know is probably something so simple im just not seeing

Answer 1

Use the fact that for any positive integer $n$, $n! = n(n-1)!$. Therefore, $$(t-1)! = (t-1)(t-2)!,\\ (n-1)! = (n-1)(n-2)!,$$ and so forth.