Permutation Definition, Formula, 4 Types & Examples

What Is a Permutation?

The term permutation refers to a mathematical calculation of the number of ways a particular set can be arranged. Put simply, a permutation is a word that describes the number of ways things can be ordered or arranged. With permutations, the order of the arrangement matters. There are three different types of permutations, including one without repetition and one with repetition. Permutations are different from combinations, where data is chosen from a group and the order doesn't matter.

Understanding Permutations

Permutations are concepts used in mathematics. They represent a variety of arrangements that can be possible in a group. Order is very important when it comes to permutations. This sets it apart from a combination, which is a concept where order doesn't matter. To some degree, permutations are a form of ordered combinations. We discuss combinations in a little more detail below.

There is a way you can calculate permutations using a formula. This formula is: 

P(n,r) = n! ÷ (n-r)!

where

• n = total items in the set;
• r = items taken for the permutation;
• "!" denotes taking the 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.

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 zero (0) through nine (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

Order matters here because a permutation produces the number of digit entryways rather than a combination.

Permutations vs. Combinations

Both permutations and combinations involve a group of items. For permutations, the order of the data matters. Consider the order for the combination for a safe. You have to have the order right in order to open it. As such, it must be entered exactly as scripted or it won't work.

A confusing example, though, since safe combos aren't actually combinations. Combinations don't rely on ordering or sequencing, which means the data in a group can be ordered in any way, even randomly. Having said this, there is no intention when it comes to setting combinations. They are totally random. Think of choosing items from the lunch menu at your local diner for your meal.

Another key difference between permutations and combinations is the type of data. Permutations rely on a list of things, which is why the order matters. This can be digits, letters, or people. Combinations, on the other hand, rely on a group of things like that menu at your favorite diner. That's why the order doesn't matter at all. So this can mean a selection of people from a sports team or the selection of items from a restaurant menu.

Types of Permutations

There are different types of permutations. The two main types of permutations are:

• Permutations with repetition. With repetition, you can make different combinations with different objects. The data is not restricted by how many times it can appear so you can use the data more than once.
• Permutations without repetition. In this case, one item is removed from the list each time you have to come up with a new permutation. Put simply, the available choices for permutations dwindle as you go along.

There are some other less common types of permutations, including permutations using multi-sets (which involves items in a list that are nondistinct) and cyclic or circular permutations or the number of ways that a number of items can be arranged around a circle.

Examples of Permutations

Here are a couple of examples to show how permutations work. These first two are from finance and business. Let's suppose a portfolio manager 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 if 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

There are plenty of examples of permutations in the real world.

• As noted above, safe combinations are actually permutations. That's because the order of the numbers matter. You can't open a safe or locker if you don't have the correct order.
• Another common example is an anagram where you make different words from the same root word. Again, order matters because you have to be able to form an actual word, not just a random sequence of letters.
• Choosing the order in which people finish a race. You can use factorials to determine the who comes in first, second, and third place, not to mention the order of the other participants.
  

The Bottom Line

Math concepts can be fairly easy to understand. A permutation is a concept that represents the arrangement of a variety of datasets from a larger list of data. And order is very important. Permutations are often confused with combinations, which are selections of data from a group of things. Permutations can be useful for financial professionals and investors alike when trying to choose investment for a portfolio.