Factorials in a Circle

Background

We know how to calculate the number of ways we can arrange distinct items (such as books each with different title) in a row (like on a shelf). Does anything change if we arrange those same items around a table?

Question

Adam, Barb, Carol, and Derek are going to play cards at a round table. The players don't care about which chair they sit in, but they do care about what order the people are around them. In how many different ways can the players sit?

Answer

(n-1)! = (4-1)! = 3! = 6

Analysis

Let's start with looking at the players, ABCD, can stand in a row (perhaps they are waiting outside the venue in order to play cards). From the Factorials entry, we know that there are 4! = 24 ways for them to stand in a row.

Now they approach the table they will play at. A will sit down first. Does it matter which chair he sits in? No - they are all equal. Where A sits has no bearing on anything.

B will now sit and gets to decide she'll sit on A's right, left, or straight across. So B has three choices as to where to sit.

C will then sit and can pick from the remaining two spots.

And then D sits in the last remaining spot.

And so the number of possible ways for the four people to sit is 3 x 2 x 1 = 6.

Remember that when we multiply natural numbers up to a given number, that's a factorial calculation that we can symbolize using an exclamation symbol. So in this case we have:

3! = 3 x 2 x 1 = 6

And in general, when we seat people at a round table, the number of ways we can seat them, looking only at the relationships of the people to each other, we can say that the number of ways they can be arranged is:

(n-1)!

Vocabulary used:

For more information check out these links (comment to add your favourite link):

Where might you have come from?

Fact-orials Index

Combinatorics:

• Factorials

Where might we go?