How does one go about upper bounding the following expression/what is the tightest upper bound one can achieve?
$$\prod_{i = 1}^k A_i!$$ such that $$\sum_{i}^k A_i \leq C$$.
What is the tighest upper bound for the first expression in terms of C and k?
Note that the trivial upper bound for this expression is $((\frac{C-k}{e})^C)^k$ but can we do better than this?