1. ## Round Table

There are 8 chairs around a round table. If A and B must sit together, then in how many ways A,B,C,D and E can sit around this table?

According to me : 4!.2! but I think I am forgetting something.

2. ## Re: Round Table

n persons can be made to sit on a round table is given by (n-1)! and two persons can be made to sit together in 2! ways. I am sure now you can reason out.

3. ## Re: Round Table

Yes but there are 8 chairs. AB must sit together. This is 4!. They can be arranged beyween themselves. This is 2!.But something is missing.

4. ## Re: Round Table

Originally Posted by kastamonu
There are 8 chairs around a round table. If A and B must sit together, then in how many ways A,B,C,D and E can sit around this table?
According to me : 4!.2! but I think I am forgetting something.
What makes this problem different is the eight chairs but only five people.

Seat A anywhere at the table. Now the tabled is ordered.
There are two places to seat B, on A's right or left.
There are then six places to seat C.
There are five places to seat D.
There are four places to seat E.

5. ## Re: Round Table

Because this is a "round table" and shifting every one to the left or right gives the same "ordering" we can set A anywhere at the table. Once we have done that, we can seat B on A's left or right- 2 choices. There are now 6 chairs left. We can seat C in any of those 6 chairs, then D in any of the 5 chairs left, then E in any of the 4 chairs left. There are 6(5)(4)= 120 ways to do that so 6(120)= 720 ways to seat these 4 people, with A and B sitting together. More formally, 6(5)(4)= 6!/3! so we could also write this as (2)(6!/3!).

6. ## Re: Round Table

But why did you multiply 120 with 6? How did you get 6?

7. ## Re: Round Table

Originally Posted by kastamonu
But why did you multiply 120 with 6? How did you get 6?

Seat A anywhere at the table. Now the tabled is ordered.
There are two places to seat B, on A's right or left.
There are then six places to seat C.
There are five places to seat D.
There are four places to seat E.

$2\cdot 6\cdot 5\cdot 4$

8. ## Re: Round Table

Many Thanks Plato. But why did HalsofIvy multyiply 120 by 6. This is what I couldn't make out.

9. ## Re: Round Table

Originally Posted by kastamonu
Many Thanks Plato. But why did HalsofIvy multyiply 120 by 6. This is what I couldn't make out.
That was clearly a typo. If you notice that his alternate way does give the correct answer.

Many Thanks.