= Solution
The <rule of product> says that if a construction has $r$ stages, with exactly $a_j$ choices at stage $j$ after every permitted earlier history, then it has $\prod_{j=1}^r a_j$ possible outcomes. The later choices need not be identical <sets> after different histories; their numbers must have the stated values.
For a <permutation> of $n$ distinct objects, choose the first object in $n$ ways, the second in $n-1$ ways, and continue until only one remains. Thus the number is
$$
\boxed{n(n-1)\cdots1=n!.}
$$
Every ordered arrangement is produced exactly once by its successive choices, so this counts <permutations> without omissions or repetitions. For $n=0$, the unique empty arrangement agrees with $0!=1$.
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