Is there a way or a general formula to write down all factorizations of $n!\ ( ∀ n ∈ N)$ into the factors which themselves are in factorial ($m!$) and combination ( $C(r,k)$ ) forms?
And also is there a formula to calculate the total number of these different factorizations for each $n!$ ?
Please note that I want a formula which has two conditions; one for $(r,k ≤ n)$, and the other for $(r,k > n)$.
$∀ n ∈ N, n! = m! ×\dots × C(r,k) × \dots$
For example:
$$ \begin{align} 8! = 2! × 3! × 5! × C(8,6)\\ 8! = 2! × 3! × C(8,6) × C(10,3)\\ 8! = 2! × 3! × 3! × C(8,4) × C(8,7)\\ 8! = 3! × 4! × C(7,4) × C(8,7)\\ 8! = 3! × 5! × C(8,3)\\ 8! = \dots\\ ... \end{align} $$
and so on.