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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.

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  • $\begingroup$ If $r > n/2$, then your formula gives zero. For instance, if $n = 3$ and $r = 2$, your formula gives $3\binom{1}{2} = 0$. $\endgroup$ Nov 11 '17 at 19:30
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    $\begingroup$ Why would placing the objects on a circle affect the number of ways of selecting the objects? $\endgroup$
    – Jens
    Nov 12 '17 at 0:00
  • $\begingroup$ Your formulas do not make sense. Please check the wording of the question. $\endgroup$ Nov 12 '17 at 10:23
  • $\begingroup$ You are right. I am editing. $\endgroup$
    – Maverick
    Nov 12 '17 at 11:29
  • $\begingroup$ In my response to this question, I explain how to derive a formula for the number of selections of $k$ objects from $n$ objects arranged in a circle if no two of the $k$ objects are consecutive. With a bit of algebraic manipulation (using Pascal's Identity) of the formula I derived, you can derive the formula $$\frac{n}{k}\binom{n - k - 1}{k - 1}$$ and then show it is equivalent to the formula $$\frac{n}{n - k}\binom{n - k}{k}$$ $\endgroup$ Nov 12 '17 at 22:05
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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.

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