What Is a Permutation?

A permutation is the way a set can be arranged, and where the order of 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 arꦆrangement is irrelevant.

Formula

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, where all the possible permutations are written.

P(n,r) = n! ÷ (n-r)!

where

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

Imagine the number of ways a sequence of a three-digit keypad can be arranged. Using the d🔴igits 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 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. Combinations don't rely on ordering or sequencing, which means the data in a group can be ordered in any way, even randomly.

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.

Types of Permutations

• Permutations with repetition: With repetition, different combinations can be made with different objects. The data is not restricted by how many times it can appear, so the data can appear more than once.
• Permutations without repetition: One item is removed from the list each time to come up with a new permutation. Put simply, the available choices for permutations dwindle as choices end.

Permutations in Finance

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 wa💖ys 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 manꦑager to construct their fund.

Suppose a c🧸ompany 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

The Bottom Line

A permut🎐ation is a concept that represents the arrangement of a variety of datasets from a larger list where order is important. Permutations can be♒ used by financial professionals and investors who choose investments for a portfolio.