On the number of combinations without certain separations

In this paper we enumerate the number of ways of selecting kk objects from nn objects arrayed in a line such that no two selected ones are separated by m−1,2m−1,…,pm−1m−1,2m−1,…,pm−1 objects and provide three different formulas when m,p≥1m,p≥1 and n≥pm(k−1)n≥pm(k−1). Also, we prove that the number of ways of selecting kk objects from nn objects arrayed in a circle such that no two selected ones are separated by m−1,2m−1,…,pm−1m−1,2m−1,…,pm−1 objects is given by nn−pk(n−pkk), where m,p≥1m,p≥1 and n≥mpk+1n≥mpk+1.