Example 1 How many different mappings are there from an $n$-element set $A=\left\{a_{1}, a_{2}, \cdots, a_{n}\right\}$ to an $m$-element set $B=\left\{b_{1}, b_{2}, \cdots, b_{m}\right\}$?
The image of $a_{1}$ can be any one of $b_{1}, b_{2}, \cdots, b_{m}$, i.e., there are $m$ choices, the image of $a_{2}$ also has $m$ choices, $\cdots$, the image of $a_{n}$ also has $m$ choices. According to the multiplication principle, there are $m^{n}$ mappings from set $A$ to set $B$.