For the set $A=\{a, b, c, d\}$, how many mappings from $A$ to $A$ are there? How many of these are one-to-one mappings from $A$ to $A$?
Solution (1) Any mapping from $A$ to $A$ can be represented as $f: a \rightarrow x_{1}, b \rightarrow x_{2}, c \rightarrow x_{3}, d \rightarrow x_{4}$, where $x_{1}, x_{2}, x_{3}, x_{4} \in B \subseteq A$. Therefore, the number of mappings $f$ from $A$ to $A$ equals the number of permutations of $a, b, c, d$ in the positions of $x_{1}, x_{2}, x_{3}, x_{4}$ with repetition allowed, which is $4^{4}=256$;
(2) The number of one-to-one mappings from $A$ to $A$ is exactly the number of permutations of $a, b, c, d$ in the positions of $x_{1}, x_{2}, x_{3}, x_{4}$ without repetition, which is $P_{4}^{4}=24$.