 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 isEvery 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 $0!=1$.