discrete mathematics - Number of bijective functions (two finite sets)

Let $M$ and $N$ be finite sets, with $|M|=m$ and $|N|=n$.

a)Find out the number of bijective functions $f: M \rightarrow N$. Look at the cases: $m \neq n$ and $m=n.$

Bijective: Every element in $N$ has exactly one partner in $M.$

$m\neq n:$ That means either $m < n $ or $m>n$.

If $m>n$ then wouldn't every element in $N$ have exactly one partner in $M$?

If $m then it wouldn't work, since some elements in $N$ wouldn't have a partner in $M$.

$m=n$: Here every element in $N$ will have exactly one partner in $M$.

So you can get a bijective functions if $m>n$ or $m=n$.

For $m=n$ number of bijective functions would be: $m$

For $m>n$ number of bijective functions would be: $n$

I'm not too sure on my answer.