KISS: Is there anything I could do with
$$ {xN\choose yN}$$
Given any size $N$, I would like to see how many ways there are to choose a fraction $yN$ out of $xN$. In factorials, that is
$$\frac{(xN)!}{(yN)![(x-y)N]!}$$
I have never seen any $(ab)!$ factorial rule, nor is there any to my intuition. Is there?
To my knowledge, there is no way to express $(ab)!$ as something else in an exact manner. However for large numbers, Stirling's approximation of the factorial can be useful, if the sought result is numerical. The Stirling approximation reads $$ n!\approx n^n\mathrm e^{-n} \sqrt{2\pi n}.$$ It is more handable in its logarithmic form $$\ln n!\approx n\ln n-n+O(\ln n).$$ The last expression is extensively used in statistical physics.