Originally Posted by
Plato Given $\displaystyle |A|=n\le|B|=m\le|C|=k$ and $\displaystyle f:A\to B,~g:B\to C$ in order for $\displaystyle g\circ f:A\to C$ to be an injection then $\displaystyle f$ must be an injection.
There are $\displaystyle \frac{m!}{(m-n)!}$ injections $\displaystyle A\to B$.
Now for $\displaystyle g\circ f:A\to C$ to be an injection it is not necessary for $\displaystyle g$ to be injective.
However, $\displaystyle g$ must be injective on the image of $\displaystyle f,~f[A].$
How many functions $\displaystyle g:B\to C$ are there such that the restriction to $\displaystyle f[A]$ is an injection?