# Arrangement

## What is an Arrangement in combinatorics ?

An arrangement of p elements from n elements is an ordered selection of p elements from n elements.

Examples:

- {2, 3, 1} is an arrangement of 3 elements from {1, 2, 3, 4}.

- {c, a} is an arrangement of 2 elements from {a, b, c}.

## Arrangement calculation Formula

Assume we have a set of n different objects then, the number of arrangements of p objects from n is equal to,

`A_n^p = frac {n!}{(n-p)!}`

Example: E is a set of 3 digits 1, 2, 3.

The number of arrangements of 2 elements from these 3 elements set is equal to 3! /(3-2)! = 6.
These arrangements are:

1, 2,

1, 3,

2, 1,

2, 3,

3, 1,

3, 2.

## How to identify an arrangement ?

there are two important criteria in identifying an arrangement:

- Order matters

- Only a subset of the whole set is concerned

Example: how many possible trifecta bets are they in a 12 horse race ?

This is an arrangement because the order (of arrival) matters but, only a subset of horses is concerned (3 of 12).

The number of possible trifecta bets is 12!/(12-3)! = 1320

## Comparison of enumeration methods

Method | Elements concerned | Order matters ? | Formula |
---|---|---|---|

Permutation | All elements (n) | Yes | `n!` |

Arrangement | Subset of p elements from n | Yes | `frac {n!}{(n-p)!}` |

Combination | Subset of p elements from n | No | `frac {n!}{p!* (n-p)!}` |