How do you form factorial if negative $(n-k)!$?

Question

How can you form factorials if they are negative?

I know that $(n+1)! = n!(n+1)$ or $(2n+2)!=(2n)!(2n+1)(2n+2).$

But what if we got $(n-1)!$ or more general, $(n-k)!$?

Answer 1

That can be achieved with $(n - 1)! = \frac{n!}{n}$. For $(n - k)! = \frac{n!}{n (n-1)(n-2)...(n - (k - 1))}, n, k \in \mathbb Z^+$ and $n \ge k$

Answer 2

You just multiply the numbers less than $n-k$ and that would be: $(n-k)!=(n-k)(n-k-1)\cdots 2 \cdot 1$