By induction, assume that any set with n elements has n! arrangments and assume that
\begin{gather*}
A = \left \{ a_1, a_2, ... , a_n, a_{n+1} \right \}.
\end{gather*}
Notice that there are n+1 ways to choose 1 element from A and that in doing so leaves a set with n elements. Combining the induction hypothesis with the multiplication principle this gives (n+1)n! = (n+1)! possible outcomes.