On the number of combinations without certain separations

Abstract

In this paper we enumerate the number of ways of selecting $k$ objects from $n$ objects arrayed in a line such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects and provide three different formulas when $m,p\ge 1$ and $n\ge pm(k-1)$. Also, we prove that the number of ways of selecting $k$ objects from $n$ objects arrayed in a circle such that no two selected ones are separated by $m-1,2m-1,...,pm-1$ objects is given by $\frac{n}{n-pk}\left(\genfrac{}{}{0ex}{}{n-pk}{k}\right)$, where $m,p\ge 1$ and $n\ge mpk+1$.