Simplifying Expression Factorial Expression I'm confused as how I'm meant to simplify this:$$\frac{(n-2)!}{(n-2-r)!}$$
I have other factorial questions where the variable isn't present in the top factorial like the question above and I'm trying to figure out how I simplify.
Thanks

\begin{align*}
\frac{(n-2)!}{(n-2-r)!} &=\frac{(n-2)!}{(n-(2+r))!} \\
&= \frac{(n-2)(n-(2+1))\dotsm(n-(2+(r-1)))(n-(2+r))!}{(n-(2+r))!}\\
&=(n-2)(n-3)\dotsm(n-1-r)\\
&=\prod_{k = 0}^{r-1}(n-(2+k))\\
&=\prod_{k=0}^{r-1}(n-2-k)
\end{align*}
For example, let $r = 3$. Then
\begin{align*}
\frac{(n-2)!}{(n-2-3)!}&=\frac{(n-2)(n-3)(n-4)(n-5)!}{(n-5)!}\\
&=(n-2)(n-3)(n-4)
\end{align*}
Using the formula
$$\prod_{k = 0}^{3-1} (n-2-k) = \prod_{k=0}^2(n-2-k) = (n-2-0)(n-2-1)(n-2-2) = (n-2)(n-3)(n-4).$$
