$n$ students are standing in a circle. Find the number of ways in which $k$ students can be selected such that no two of them are together.

Question

The question is

I know one method to do this by comparing it with a question like "find number of triangles that can be formed inside an $n$-sided polygon such that no side of triangle is common to the $n$-sided polygon" but that'll be valid only when $k=3$.

I want to know general method to solve such questions and also the principle behind that method.

Any help appreciated

Does this answer your question? No. of polygons in a polygon with no side coinciding. — reyna, Oct 25 '20 at 04:54
It will be easier to understand with a specific value for $k$. Check out Counting heptagons formed using vertices of n-sided polygon having no sides in common with the polygon — reyna, Oct 25 '20 at 04:57
@zaira Thanks for sharing those but I am having a tough time understanding them. I'd like if someone writes a more elaborated answer on this.Doesn't matter if he/she solves it for a particular values of n and k — Ashu tosh, Oct 25 '20 at 05:08
@N.F.Taussig yes that's helpful too!Thanks for sharing.Though I am having problem in understanding the "solution to the recurrence part(the obvious part :/ )" . His final result is giving me the correct answer that I require ,so his question answers what I want. I'll try to understand again after sometime maybe I'll get it then..Also,Thank you for the Edit. — Ashu tosh, Oct 25 '20 at 12:34
@Ashutosh Welcome to MathSE. This tutorial, which I forgot to tell you about earlier, explains how to typeset mathematics on this site. — N. F. Taussig, Oct 25 '20 at 12:55
There is another approach to the same problem in this question. — N. F. Taussig, Oct 25 '20 at 12:58
@N.F.Taussig by the way ,are u familiar with method for selection in which we take approach as :- Taking varibales $x,y,z,w etc$ and then putting $x+y+z+w=n$ and, since all these are integers ,we write their maximum and minimum values. And then proceed for answer by "Coefficient of $\alpha^k$ in ($\alpha^2 + \alpha^3$ +... ).. If you are familiar with this approach ,can u help me solving this problem using this as this is the method I'll be able to grasp easily — Ashu tosh, Oct 25 '20 at 13:04
I didn't get my required answer and you guys closed my question — Ashu tosh, Oct 25 '20 at 15:16
You may find Brian M. Scott's addendum to his answer in this question useful. — N. F. Taussig, Oct 25 '20 at 17:43

$n$ students are standing in a circle. Find the number of ways in which $k$ students can be selected such that no two of them are together.

0 Answers0