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.