Arrangement

Calculates the number of p-element arrangements from a set of n elements (n >= p).

p ≤ n


This online tool calculates the number of p-element arrangements for a set of n elements. In combinatorics, an important criteria in identifying an arrangement is that the order of the elements matters.

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)!}`

See also

Permutation
Combination