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 n n − p k ( n − p k k ) , where m , p ≥ 1 and n ≥ m p k + 1 .