Number of ways to select $r$ objects from $n$ objects placed on the circumference of a circle Prove using combinatorial argument that number of ways to select $r$ objects from $n$ distinct objects such that no two selected objects are consecutively placed on the circumference of a circle is $$\frac{n}{n-r} \binom{n-r}{r}.$$
I am aware that number of ways to do so is $$\frac{n}{r}\binom{n-r-1}{r-1}.$$ But I would want to understand how $$\frac{n}{n-r} \binom{n-r}{r}$$ comes combinatorially. I think it has something to do with menage problem.
 A: Let the objects be placed on the vertices of an $n$-gon. You ask how many ways are there to select $r$ of the objects such that no two selected objects are consecutive.
Your problem is equivalent to the problem of placing $r$ non-overlapping dominoes on the vertices of an $n$-gon. (Each domino covers two vertices.) To map the domino problem to your problem, let the object sitting on vertex $i$ be one of the selected ones. Then place one of the dominoes so that it covers vertices $i$ and $i+1$, with $i+1$ computed mod $n$. The condition that no two selected objects be consecutive is equivalent to the condition that the dominoes be non-overlapping.
We may then apply the solution given in my answer here to the domino formulation of the problem. Briefly, let $D$ represent a domino and let $V$ represent a bare vertex (one not covered by a domino). There are $r$ $D$s and $n-2r$ $V$s on our circle. The factor $\binom{n-r}{r}$ in your expression is the number of words one can make with these letters. Each such word may be wrapped on the circle in $n$ ways. But this overcounts domino configurations by a factor of $n-r$ since each of the $n-r$ cyclic permutations of the letters of a word gives rise to the same domino configurations.
Incidentally, the connection with the ménage problem is that the expression $\frac{2n}{2n-r}\binom{2n-r}{r}$ appears in the Touchard formula for the number of configurations:
$$
2\cdot n!\sum_{r=0}^n(-1)^r\frac{2n}{2n-r}\binom{2n-r}{r}(n-r)!.
$$
Domino configurations have been used to enumerate unwanted configurations that need to be excluded in an inclusion-exclusion argument. See this article.
