A permutation of a set of distinct objects is an ordered arrangement of these objected.

For a set of \(n\) objects, there exists \(n!\) permutations of these.

Note: this is different to an unordered arrangement, which are Combinations.

## k-permutations

To find permutations of \(k\) objects within a set of \(n\), a *k-permutation* can be calculated.

Given \(n, k\) where \(0 \leq k \leq n\). There are
\[
P(n,k) = n(n-1)\ldots(n-k+1) = \frac{n!}{(n-k)!}
\]
This may also be seen denoted as \(P^n_k\), \(_nP_k\), \(^nP_k\) and other arrangements, depending on author/context.

## With repetition

Cases where there is a need to find \(r\) permutations of a set of \(n\) objects where repetition is allowed are simply \(n^r\).

Concretely, if we consider a string of length 5 made of lowercase letters, there are 26 options for first, 26 for the second and so on, giving \(26 \times 26 \times 26 \times 26 \times 26 = 26^5\). This case mirrors the counting product rule.