Permutations In a Circle

September 22, 2008

There are a lot of varieties of permutation problems (permutations of permutations, perhaps?). Most of them fall into predictable patterns, and you don't need a lot of tools to attack them.

However, there's one type that always gives people trouble: When items are arranged in a circle.

Here's an example, taken from my Extreme: Challenge problem set:

In a garden, seven flowers are to be arranged around a circular walk. Two arrangements of the flowers are considered different only when the positions of the flowers are different relative to each other. What is the total number of different possible arrangements of the flowers?

Even if you've never seen a problem like this before, you might suspect that something is different that your typical permutation problem. All that verbiage about "different relative to each other" doesn't come up in other settings.

Circular Permutations and Row Permutations

If the seven flowers were to be arranged in a row, this would be a relatively simple question. There would be seven flowers to choose from in the first position (say, on the left-hand side), then six for the next, then five, and so on. The answer would be 7!, or 5,040.

It's a safe bet, then, that the circular scenario differs. Consider the following two "rows" of seven flowers, in which each different flower is labeled with a different letter:

• A B C D E F G
• B C D E F G A

In a "row" question, those are two different permutations. However, if you arrange them in a circle, the pattern--A B C D E F G--is the same. The positions of the flowers are not different relative to each other.

So, just how different are these scenarios?

Double-Counting Circular Permutations

In this case, there are seven different rows that end up being the same circular arrangement:

• A B C D E F G
• B C D E F G A
• C D E F G A B
• D E F G A B C
• E F G A B C D
• F G A B C D E
• G A B C D E F

For any "row" arrangement you can think of, there are six other row arrangements that result in the same circular arrangement. Thus, each circular arrangement represents seven different rows.

Putting that into practice, you can start by finding the number of row arrangements: 7!. Since that is 7 times the correct answer, divide it by 7:

7!/7 = 6! = 720

The General Solution

Just as "row" permutations (at least, problems without further wrinkles) can be solved by finding n!, circular permutations are (n-1)!

That doesn't mean I've just given you a simple fact to memorize. Few GMAT questions will be this simple. It's important to understand all the underlying steps discussed here, so that if an additional twist is added to a circular permutations question, you know how to apply this technique, even if it isn't as simple as (n-1)!.

About the author: Jeff Sackmann is a GMAT tutor based in New York City. He has created many resources for GMAT preparation, including the popular Total GMAT Math and Total GMAT Verbal, as well as 1,800 practice GMAT math questions.

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