What Is a Permutation?

A permutation is a mathematical calculation of the number of ways a particular set can be arranged, where the order of the arrangement matters. 

Formula and Calculation of Permutation

The formula for a permutation is: 

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?" A permutation can be calculated by hand as well, where all the possible permutations are written out. In a combination, which is sometimes confused with a permutation, there can be any order of the items.

Key Takeaways

  • Premutation is the number of ways a set can be arranged. 
  • Roughly, it means, “how many ways can something be arranged.”  
  • The order of numbers in a permutation, with a combination, however, the order does not matter. 

What Permutation Can Tell You

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 entryways, 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 example would be, 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.

Permutations vs. Combinations

Both permutation and combinations involve a group of numbers. However, with permutations the order of the numbers matters. With combinations, the ordering does not matter.  For example, with permutation, the order matters, such as the case with a locker combination. 

Locker combos are, thus, not combinations. They are permutations. A locker combo must be entered exactly as scripted, such as 6-5-3, or it will not work. If it were a true combination then the numbers could be entered in any order and work. 

There are various types of permutations as well. You can find the number of ways of writing a group of numbers. But you can also find permutations with repetition. That is, the total number of permutations when the numbers can be used more than once or not at all. 

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