Let's say we're asked to arrange these factorials in descending order: $1000!, \,\,700!\cdot300!,\,\,500!\cdot500!,\,\,600!\cdot300!\cdot100!$
For the first three, we could do:
$\dbinom{1000}{300}>1 \Rightarrow 1000!>700!\cdot300!$
$\dbinom{1000}{500}>\dbinom{1000}{300} \Rightarrow 700! \cdot 300! > 500! \cdot 500!$
However I don't know how to approach the fourth one, i.e how do we prove that $500!\cdot500!>600!\cdot300!\cdot100!$ ?
What is $$ \frac{(a+b+c)!} {a! \,b! \,c!} $$