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Permutations

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.