# Find the number of ways to choose $k$ objects from a set of $n$ objects arranged in a circular order such that no two consecutive elements are chosen

Find the number of ways to choose $$k$$ objects from a set of $$n$$ objects arranged in a circular order such that no two consecutive elements are chosen.

I could think of a combinatorial solution-

Arrange the objects such that they are in a row and not in a circular order. To begin with, the objects are arranged as $$\{1,2,3,\cdots,n\}$$ (in that order). From the stars and bar method it can be seen that there are a total of $${n-k-1}\choose{k-1}$$ ways to choose $$k$$ objects from the row such that no two are adjacent assuming that object $$1$$ is chosen. This can be repeated and we can start from all $$n$$ objects instead of object $$1$$. But then there's going to be $$k$$ repeats. And thus the answer gets boiled down to $${\frac{n}{k}} \times {{n-k-1}\choose{k-1}} = \left(\frac{n}{k}\right) {{n-k-1}\choose{k-1}}$$.

I have this solution but then I feel that there are some nice ways using generating functions to solve this. Can anyone help me out?