On the number of combinations without certain separations

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 $m-1,2m-1,\mathrm{.}\mathrm{.}\mathrm{.},pm-1$ objects and provide three different formulas when $m,p\geq 1$and$n\geq pm(k-1)$\mathrm{.}Also,weprovethatthenumberofwaysofselecting$k$objectsfrom$n$ objects arrayed in a circle such that no two selected ones are separated by $m-1,2m-1,\mathrm{.}\mathrm{.}\mathrm{.},pm-1$ objects is given by $\frac{n}{n-pk}\left(\genfrac{}{}{0px}{}{n-pk}{k}\right)$, where $m,p\ge 1$ and $n\ge mpk+1$.