Gossiping Models Formal Analysis of Epidemic Protocols
for any n ≥ 0 and 0 ≤ k ≤ n. A combinatorial interpretation of the formula (1) is as follows. The left-hand side counts the number of ways to select k balls (numbered with the corresponding number) from a bin of n balls. This is equivalent to the sum of the following two cases: (a) if the ball n is selected, the number of ways to select the remaining k − 1 balls from n − 1, and (b) if the ball n is not selected, the number of ways to select all k balls from n − 1. This is often a good approach to understand binomial coefficient identities. The interpretation above is considered as a combinatorial proof of the identity (1) as it gives two different solutions to the same counting problem. The combinatorial identities and, in particular, the identities involving binomial coefficients have been studied in, e.g., [5, 9, 13, 15, 8, 12]. In this paper, we describe some identities involving binomial coefficients that are the result of a non-trivial counting problem. This paper is organized as follows. Section 2 describes the counting problem that we are interested in. Section 3 presents three answers to the stated problem. The first and the third answer are exact solutions to the counting problem, thus establishing a combinatorial proof of their equality. The second answer has been obtained by extensive experiments, its correctness was hitherto not proven. In Section 4, we give an algebraic proof of a identity of the second and the third solution, thereby proving its correctness and the equality of all solutions. Section 5 concludes the paper.