Minimizing a product of factorials

Question

Im trying to minimize $n_1! n_2! \cdots n_k!$ under the constraint $n_1 + n_2 + \cdots + n_k = N$ and $n_i > 0$ for all $i$. I know the answer is that it is minimized if all $n_i$ are equal to $\frac{N}{k}$ for all $i$. There is already a post on this forum about maximizing and apparently one answer there is given for minimizing. But i’m having a very hard time understanding it since this is my first time seeing gamma and betta functions. I tried to use induction but i got stuck there too. I’m a first year math major.

Answer 1

Suppose $n_1, \ldots, n_k$ satisfies the constraint and $n_{\max} - n_{\min} > 1$. Then increasing the smallest $n_i$ by one and decreasing the largest $n_i$ by one leads to another configuration that satisfies the constraint. Furthermore this changes the product of factorials by a factor of $\frac{(n_{\min}+1)! (n_{\max}-1)!}{n_{\min}! n_{\max}!} = \frac{n_{\min}+1}{n_{\max}} < 1$. Therefore, the original configuration was not the minimizing choice.

Thus, the minimizing choice must satisfy either $n_{\max}-n_{\min}=1$ or $n_{\max}-n_{\min}=0$. If $N$ is divisible by $k$, then the latter is possible; otherwise the best one can do is the the former case with a configuration consisting of $n_i$ being a combination of $\lfloor N/k\rfloor$ and $\lceil N/k \rceil$.