Permutations With Repetition

A permutation with repetition is an ordered arrangement of $k$ items chosen from a set of $n$ distinct objects, where the same object may be selected more than once.

Since each of the $k$ positions can be filled independently by any of the $n$ available objects, the total number of such arrangements is

$$n^k.$$

For example, the number of $2$-letter codes formed from the letters $\{A, B, C\}$ with repetition allowed is $3^2 = 9{:}$

$$AA,\ AB,\ AC,\ BA,\ BB,\ BC,\ CA,\ CB,\ CC.$$

Notice that order matters: $AB$ and $BA$ are counted as different arrangements.