Suppose that you wanted to take the ratio of two negative factorials, such as: $\frac{(-n)!}{(-m)!}$ for positive integers $n,m>0$. I am pretty sure it is well defined (binomials with a negative upper index), however I do not know how to show that through factorials.
The best shot I have at it is: $$ \frac{(-n)!}{(-m)!}=\left (\frac{\pi n}{\sin \pi n} \right ) \frac{1}{n!} \left(\frac{\sin \pi m}{\pi m} \right ) m! = \frac{(m-1)!}{(n-1)!}\frac{\sin \pi m}{\sin \pi n} $$ which has different answers (zero, infinity, or $\frac{(m-1)!}{(n-1)!}$ ) depending on the order of which limits are taken (with respect to $m$ or $n$).
The factorials of negative integers can be related to the simple poles of the gamma function by:
$$ m! = \Gamma(m+1) $$
The gamma function is meromorphic, and the behavior of $\Gamma(z)$ as $z$ approaches a non-positive integer $-m+1$ is like $R/(z+m-1)$ with residue:
$$ R = \frac{(-1)^{m-1}}{(m-1)!} $$
Therefore the most natural interpretation for positive integers $n,m$ is:
$$ \frac{(-n)!}{(-m)!} = \frac{\Gamma(-n+1)}{\Gamma(-m+1)} = (-1)^{n-m}\frac{(m-1)!}{(n-1)!} $$
This is equivalent to letting complex arguments approach $-n,-m$ at equal rates.