How many maps can exist between two sets?

Question

I'm working on the following exercise. Why does the solution omit applying induction on $n$? That is, assume $P(n)$ and then use that assumption to prove $P(n + 1)$.

Answer 1

As pointed out by @RobertIsrael in the comments, in this particular problem, it's easier to do the induction on $m$ because it's easy to assign an extra element in $E$ to an element in $F$ while it's not that straightforward the other way around.

I must admit though, this problem can be easily solved without the help of induction. Note that you can assign any of the $m$ elements of $E$ to any of the $n$ elements of $F$. That is, there are $n$ choices for each of the $m$ elements. As the choices are independent, the total number of possible maps is $n^m$.

EDIT: As noted in the comments, the alternative method does include induction and is actually similar to the previous logic. Just that the induction is not explicitly used. The combinatorial argument used is based on induction.