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, 2 m - 1, ..., p m - 1 objects and provide three different formulas when m, p ≥ 1 and n ≥ p m (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, 2 m - 1, ..., p m - 1 objects is given by frac(n, n - p k) (frac(n - p k, k)), where m, p ≥ 1 and n ≥ m p k + 1.