There are 18 chairs numbered from 1 through 18 around a circular table. How many ways can three people be seated, so that no two people are adjacent?

ramenmaster28 Jan 20, 2020

#1**+1 **

We can use a stars-and-bars technique. First, we cut the circle at a certain point, to get a row of chairs. Using stars-and-bars, there are C(16,3) = 560 ways to choose two chairs so that no two chairs are adjacent. Therefore, there are 560*6 = 3360 ways to seat the three people.

Guest Jan 20, 2020

#2**-2 **

I tried this before an got in a mess. This time I am more confident.

But there is still plenty of room for error.

Seat each person and tie their chair to the chair on the right (which will be empty.)

Think of the 2 chairs and the one person as a single unit.

So now you have 3 fat units and 12 more chairs.

You have to slip the fat units between the other chairs.

15 units altogther and you need to choose the 3 that are fat.

so that is 15C3= 455

Now the three people can be in any order so that is 3!=6 ways

455*6=2730 ways

Melody Jan 21, 2020