Use this online tool to calculate the number of permutations of a set of n elements. In combinatorics, the order of the elements counts in a permutation.

What is a permutation ?

A permutation is an ordered arrangement of a set of objects. More precisely, if we assume we have a set of n elements (elements may be letters, digits, objects), a permutation is any possible arrangement of these n elements. Note that order matters in a permutation.

- {2, 3, 4, 1} is a permutation of {1, 2, 3, 4} set.
- {2, 3, 1, 4} is another permutation of {1, 2, 3, 4} set.
- {b, c, a} is a permutation of {a, b, c}.
- {c, b, a} is another permutation of {a, b, c}.
- {B, A} is not a permutation of A, B, C because we didn't keep all the elements (C is missing).

Calculation Formula

For a set of n different objects then, the number of permutations of that set is equal to,

`P_n = n!`

Example : E is a set of 3 elements which are 1, 2, 3.
The number of permutations in this set is equal to 3! = 6.
The permutations of E are :
1, 2, 3,
1, 3, 2,
2, 1, 3,
2, 3, 1,
3, 1, 2,
3, 2, 1.

How to identify a permutation ?

There are two important criteria in recognizing a permutation,
- order matters
- the whole (not a subset) of the elements is concerned

How many possible rankings in a 12 horse race ?
This is a permutation because the order (of arrival) matters and all the horses are concerned.
The number of possible rankings is 12! = 479 001 600.

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