*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: *n*^{k}.