# Round Table Problem

• Apr 12th 2012, 12:12 PM
jpicks23
Round Table Problem
There are six chairs at a round table. A seating arrangement will be considered the same if everyone at the table has the same neighbor to the left and to the right.

a) How many different ways can we seat six people, assuming that one person sits in each chair?
b) How many different ways can we seat four people, assuming that two chairs are left empty?
c) How many different wats can we seat three couples if every person is seated next to their date?

a) is pretty obvious, which is 5!. However I'm not quite sure how to approach b and c. Any ideas? Thanks
• Apr 12th 2012, 12:39 PM
Plato
Re: Round Table Problem
Quote:

Originally Posted by jpicks23
There are six chairs at a round table. A seating arrangement will be considered the same if everyone at the table has the same neighbor to the left and to the right.
a) How many different ways can we seat six people, assuming that one person sits in each chair?
b) How many different ways can we seat four people, assuming that two chairs are left empty?
c) How many different wats can we seat three couples if every person is seated next to their date?
a) is pretty obvious, which is 5!. CORRECT

How many ways can we have two empty chairs at a round table?
In is not three: next to each other, one chair apart, two apart?
Now the table is ordered. Seat the four in the non-empty seats?

How many ways can couple A be seated at the table? Is it now ordered?

• Apr 12th 2012, 01:21 PM
jpicks23
Re: Round Table Problem
Quote:

Originally Posted by Plato
How many ways can we have two empty chairs at a round table?
In is not three: next to each other, one chair apart, two apart?
Now the table is ordered. Seat the four in the non-empty seats?

How many ways can couple A be seated at the table? Is it now ordered?

There're 6 ways you can put 2 empty chairs next to each other, 6 for a chair apart, and 6 for 2 chairs apart. Anything past 2 will be repetitive. But now how do you determine the number of ways the 4 people can be seated? Is it (6 x 4!) + (6 x 4!) + (6 x 4!)?

As for c), would 3! = 6 be correct?
• Apr 12th 2012, 01:26 PM
Plato
Re: Round Table Problem
Quote:

Originally Posted by jpicks23
There're 6 ways you can put 2 empty chairs next to each other, 6 for a chair apart, and 6 for 2 chairs apart. Anything past 2 will be repetitive. But now how do you determine the number of ways the 4 people can be seated? Is it (6 x 4!) + (6 x 4!) + (6 x 4!)?

NO indeed. There are only three ways to pick the two empty chairs period.
• Apr 12th 2012, 03:02 PM
jpicks23
Re: Round Table Problem
Quote:

Originally Posted by Plato
NO indeed. There are only three ways to pick the two empty chairs period.

So it's 3 x 4! for the total number of ways to seat 4 people, correct?
• Apr 12th 2012, 03:12 PM
Plato
Re: Round Table Problem
Quote:

Originally Posted by jpicks23
So it's 3 x 4! for the total number of ways to seat 4 people, correct?

No that is not correct. The answer is 60.
Look at the diagram.Attachment 23583
Think how many ways to seat four people for each of those three cases.