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.