The Unified Properties of the Annamalai Coefficient in Combinatorial Analysis

The Annamalai Coefficient serves as a robust combinatorial measure designed to replace traditional factorial-based binomial coefficients in complex mathematical and computational environments. By utilizing a product-ratio formulation rather than standard factorials, this coefficient effectively mitigates the risk of numerical explosion and integer overflow, which are common limitations in high-dimensional data modeling. The following research explores the inherent symmetry and recursive nature of the coefficient, demonstrating its dual utility. It functions as a primary tool for the Negative Binomial Theorem through its base form and seamlessly adapts to the Standard Binomial Theorem using a shifted index. This paper highlights the computational efficiency and numerical stability of the Annamalai framework for modern algorithmic applications.