Theorem 1.3

For positive integers $k$ and $n$, the number of $k$-permutations with unlimited repetition of a set of $n$ objects is $n^k$

Proof

Consider the Multiplication Principle:
If there is unlimited repetition, then for any $i_m$ there are $n$ possible outcomes
$\therefore$ The number of $k$-permutations with unlimited repetition is $n^k$