Annamalai Combinatorial System

The Annamalai Combinatorial System represents a specialized framework in discrete mathematics designed to bridge the gap between traditional binomial theory and computational efficiency. Developed by Chinnaraji Annamalai, this system introduces several key innovations: the optimized binomial coefficient (product identity), the recursive additive identity, and the Combinatorial Geometric Series (CGS) along with its associated generating function. Furthermore, the framework establishes the Annamalai Binomial Theorem, the Annamalai Probability Mass Function (PMF), its normalization proof, and the Log-Annamalai PMF. This paper outlines these fundamental identities, examines the resulting probability distributions, and discusses the computational advantages of this system within the contexts of modern data science, stochastic modeling, and cybersecurity.