In this article we find Is (n, k, m), the number of ways of selecting k balls from n balls arranged in a line with exactly m 's-successions, which gives a generalization of (4.1) in [2]. Also, two explicit formulae (3.2) and (3.5) are obtained for Is (n, k, r, ri and ds (n, k; r, r') respectively, which denote the number of ways of selecting k balls from n balls arranged in a line and a circle with no two adjacent balls being of separation s, satisfying the condition r s r' o r, r' n. Special cases and a combinatoral identity are given.