A direct argument for Kaplansky’s theorem on a cyclic arrangement and its generalization

Kaplansky gave a formula which determines the number of ways of selecting j objects, no two consecutive, from n objects arrayed in a cycle. Possibly due to a lack of intuitive and direct argument for this result, generalization to ‘no k consecutive’ has not been obtained as it has for the line case. This paper gives such an argument and consequently generalizes Kaplansky’s result to arbitrary k≥2 in a unifying way.