What are the number of ways of arranging k out of n chairs in a circle such that no two of the k chairs are adjacent to each other?
e.g If n=4 and k =2 then number of ways is 2 .
For simplicity, say the k chairs are red, and all the other chairs are green. You want to count circular arrangements where no two red chairs are next to each other.
You know that each red chair must have a green chair on its right. Treat these pairs of red and green chairs as one object (i.e. they move around together).
So for n=4 k=2
(r1 g1)(r2 g2)
(r1 g2)(r2 g1)
Or for n=5 k=2
(r1 g1)(r2 g2)g3
(r1 g1)g3(r2 g2)
(r1 g2)(r2 g1)g3
(r1 g2)g3(r2 g1)
(r1 g3)g1(r2 g2)
...
See how it works? Think in terms of red-green pairs of chairs.
I have been reluctant to join this discussion.
The reason being that while I agree with snowtea’s assignment of ‘red’ chairs, why do we assume that the chairs are distinct?
If the chairs are not distinct, then the basic formula is incorrect.
Why do we assume the chairs are distinct apart from color?