Cyclic permutations: Ten persons sit around a circular table

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Ten persons -- A, B, C, ..., I, and J -- sit around a circular table. The chairs are all numbered.

What is the probability that A and B sit next to each other?
 
Cyclic permutations. I hope I've got this right. I trust pka will straighten me out, if not.

There are (n-1)! ways to arrange n people around a circular table.

Now, A and B must be seated next to one another. Think of them as tied together or as 1 morbidly obese person. Therefore, you have 8! possible arrangements. Also, the 2 can be arranged in 2! ways among themselves.

So, (8!)(2)=80640

The number of possible arrangements around the table are 9!=362880

80640/362880=2/9
 
Re: Cyclic

Hello, atomos!

I have a back-door approach to this one . . .

Ten people \(\displaystyle \{A,\,B,\,C,\,\cdots\,,J\}\) sit around a circular table.
The chairs are all numbered.
What is the probability that \(\displaystyle A\) and \(\displaystyle B\) sit next to each other?
Click to expand...

\(\displaystyle A\) can sit in any of the ten chairs . . . right?

Then the probability that \(\displaystyle B\) sits in an adjacent chair is: \(\displaystyle \L\,\frac{2}{9}\)

 
Hello, Cody!

Yes . . . and good work!

But when I saw your simple answer \(\displaystyle \left(\frac{2}{9}\right)\), I just had to see why.

And when I saw why, it was AH-HA! time
\(\displaystyle \;\;\)(My big chance to look like a genius . . . )

 
not circular permutation?

I think itsn't a circular permutation, since all chairs are numbered. Eventhough the table is round.
So I'm sorry if I said it was circular permutation :wink:

so the probability of A and B sit next to each other :
\(\displaystyle \frac{10.8!}{10!}=\frac{1}{9}\)
Is it right or wrong ?
thanx
 
not circular permutation?

I think itsn't a circular permutation, since all chairs are numbered. Eventhough the table is round.
So I'm sorry if I said it was circular permutation :wink:

so the probability of A and B sit next to each other :
\(\displaystyle \frac{10.8!}{10!}=\frac{1}{9}\)
Is it right or wrong ?
thanx
 
What possible difference could the fact that the chairs are numbered make?
Seat A at the table. Once that is done the table is ordered.
There are nine places to seat B, two of them are next to A: 2/9.
 
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