Solving $(\prod^t_{i=1} N_i m_i)!$

Question

I would like to know how to solve or simplify the factorial $(\prod^t_{i=1} N_i m_i)!$.

Here, $i, N_i, m_i$ are positive integers.

My effort:

$$(\prod^t_{i=1} N_i m_i)!$$ $$\implies (\prod^t_{i=1} N_i m_i) (\prod^t_{i=1} N_i m_i - 1)!$$ $$\implies (\prod^t_{i=1} N_i m_i) (\prod^t_{i=1} N_i m_i - 1) (\prod^t_{i=1} N_i m_i - 2) \ldots 3.2.1$$

I am not sure how to proceed from here.