Possible Duplicate:
Number of triangles inside given n-gon?
Let $f(n,k)$ denote the number of arrangements of $k$ stones and $n-k$ sticks in a circle in such a way that no two stones are adjacent to each other. If one arrangement can be obtained from another after a circular shift, it is considered different. I am to prove that $f(n, k) = \dfrac{n}{k}{n-k-1 \choose k-1}$, where $0 < k \leq n$.
What I've been thinking of is to begin with a chain of consecutive $k$ stones/sticks \[ \cdots \circ | \circ | \circ | \circ \cdots \]
(they're supposed to be arranged in a circle). This serves as a "base" for any valid arrangement of $k$ stones, $n-k$ sticks (because there must be at least one stick between two consecutive stones). Then we insert the remaining $n-2k$ sticks in the $k$ "separating" places (no matter whether we add a new stick immediately before or after an existing stick, as they're indistinguishable). There're ${n - k - 1 \choose k-1}$ such distributions.
I'm stuck at this point though, since this is mereley the number of different patterns of arranging sticks and stones. What I'm still missing is, how to count the number of ways to "apply" those patterns to the circle (i.e. how to fit the stones and sticks in the $n$ distinct places around it).
I believe this isn't a very difficult problem, but somehow I can't find a way to reach the desired result and need some help.