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.
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.
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.