Which is bigger $(n!)^m$ or $(m!)^n$?

Question

Assume $m>n$, which is bigger $(m!)^n$ or $(n!)^m$?

This question came about during a Taylor series approximation.

Considering plot of $(n!)^{1/n}$ and Stirling's formula one guesses larger base wins.

Answer 1

Consider $m = n+1$. Then $(m!)^n > (n!)^m$.

Proof:

$$ \begin{eqnarray*} (m!)^n &=& ([n+1]!)^n \\ &=& (n+1)^n \cdot (n!)^n \\ &=& \frac{(n+1)^n}{ (n!) } (n!)^{n+1}\\ &=& \frac{(n+1)^n}{ (n!) } (n!)^{m}\\ \end{eqnarray*} $$ Now since $n+1 > k$ for all $k = 1, 2, \ldots, n$, It is clear that $\frac{(n+1)^n}{n!} > 1$, proving the result.

A simple induction then shows if $m > n$, then $(m!)^n > (n!)^m$ in general.

Answer 2

Repeated use of the fact $n!\leq n^n$ yields

$$n!\cdot n!\cdots n!\lt (n+1)^n\cdot (n+2)^n\cdots (n+i)^n$$ for any $i\gt 0$, where there are $i$ terms on the LHS. Then we have

$$(n!)^i\lt \Big( (n+1)\cdots(n+i)\Big)^n$$

or equivalently

$$(n!)^i\lt \left( \frac{(n+i)!}{n!}\right)^n.$$

Rearranging, this says

$$(n!)^{n+i}\lt (n+i)!^n$$

so if we know $m\gt n$ then $m=n+i$ for some $i\gt 0$, so this tells us finally that

$$(n!)^{m}\lt (m)!^n$$