# Arrangement

## Arrangement

In a set E of n elements, this is an ordered subset with k elements of E without repetition.

Because an arrangement is an ordered subset, it is preferable to use the notation in the form of n-uplet to designate an arrangement.

• arrangement with repetition
In a set E of n elements, it is an ordered subset of k elements of E with the possibility of repetitions.

### Examples

Consider the set E = {0, 1, 2, 3, 4, 5}

Object A = (0, 2, 4) is an arrangement of three elements of E without repetition. Object B = (2, 0, 4) is a different arrangement from A.

Object B = (1, 1, 3, 4) is an arrangement of four elements of E with repetitions.

Let’s consider the set F = {Δ, , ◊}. The six arrangements (without repetition) of the elements of F taken two at a time are :
{(Δ, ), (Δ, ◊), (, ◊), (, Δ), (◊, Δ), (◊, )}.

### Notation and Formula

The number of arrangements in a set E that includes n elements taken k at a time is given by the formula: $$A_n^k =\dfrac{n!}{(n-k)!}$$.

The number of arrangements with repetition of a set E including n elements taken k at a time is given by the formula: nk.