DEFINITION of Permutation

Permutation is a mathematical calculation of the number of ways a particular set can be arranged, where order of the arrangement matters. The formula for a permutation is given by: 

P(n,r) = n! / (n-r)!


n = total items in the set; r = items taken for the permutation; "!" denotes factorial

The generalized expression of the formula is, "How many ways can you arrange 'r' from a set of 'n' if the order matters?" In a combination, which is sometimes confused with a permutation, there can be any order of the items.


A simple approach to visualize a permutation is the number of ways a sequence of a three-digit keypad can be arranged. Using the digits 0 through 9, and using a specific digit only once on the keypad, the number of permutations is: P(10,3) = 10! / (10-3)! = 10! / 7! = 10 x 9 x 8 = 720. In this example, order matters, which is why a permutation produces the number of digit entry ways, not a combination.

In finance and business, here are two examples. First, suppose a portfolio manager has screened out 100 companies for a new fund that will consist of 25 stocks. These 25 holdings will not be equal-weighted, which means that ordering will take place. The number of ways to order the fund will be: P(100,25) = 100! / (100-25)! = 100! / 75! = 3.76E + 48. That leaves a lot of work for the portfolio manager to construct his fund!

An easier one for the mind to grasp: Say a company wants to build out its warehouse network across the country. The company will commit to three locations out of five possible sites. Order matters because they will be built sequentially. The number of permutations is: P(5,3) = 5! / (5-3)! = 5! / 2! = 60.