Arrangement

• The number of possibilities to form an ordered subgroup with no repetitions
• Also known as
• k-permutations of n
• partial permutation
• sequence without repetition
• variation

Formula

• $n$: available number of elements
• $k$: total elements in the subgroup

$$A_{n,k} = \frac{n!}{(n-k)!}$$

Other symbols

$$A_{n,k} = P(n,k) = {n}P{k} = (n)_{k} = n ^{\underline{k}}$$

Example

• You have a set {1, 2, 3, 4} ($n = 4$) and you need to find all possible ordered subgroups of size 2 ($k = 2$), you would have the following subsets:
• 12 possibilities!

{1, 2}
{1, 3}
{1, 4}

{2, 1}
{2, 3}
{2, 4}

{3, 1}
{3, 2}
{3, 4}

{4, 1}
{4, 2}
{4, 3}

Arrangement with repetition

• An arrangement with repetitions allowed
• Also known as
• k-tuples
• permutations with repetition (although it's not a permutation in the usual sense)