K Permutations of N With Repetition

Counting the number of ordered sequences of length k drawn from a set of n distinct items, where items may be repeated. The count is n^k, generalized by the multiplication principle to handle positions with different or restricted numbers of choices.

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Tutorial

K-Permutations With Repetition

A $k$ -permutation of $n$ with repetition is an ordered sequence of length $k$ drawn from a set of $n$ distinct items, where each item may be reused any number of times. The number of such sequences is

n^k.

For example, the number of binary strings of length $3$ is $2^3 = 8.$ Listing them confirms this:

000,\ 001,\ 010,\ 011,\ 100,\ 101,\ 110,\ 111.

Order matters, so $001$ and $010$ count as distinct sequences.