The number of s-separated k-sets in various circles

This article studies the number of ways of selecting $k$ objects arranged in $p$ circles of sizes $n_1,\dots ,n_p$ such that no two selected ones have less than $s$ objects between them. If $n_i\ge sk+1$ for all $1\le i\le p$, this number is shown to be $\frac{n_1+\dots +n_p}{k}\left(\genfrac{}{}{0px}{}{n_1+\dots +n_p-sk-1}{k-1}\right)$. A combinatorial proof of this claim is provided, and some nice combinatorial formulas are derived.