The optimized binomial coefficient or binomial expansion has been derived from the multiple summations of geometric series. This binomial coefficient is a methodological advance in the areas of combinatorics and theoretical computer science. This paper presents a new theorem with binomial...

This paper presents lemmas and its corollaries on the combinatorial geometric series and summation of series of binomial coefficients. Also, the coefficient for each term in combinatorial geometric series refers to a binomial coefficient. These ideas can enable scientific researchers to solve...

This paper presents the relations of optimized combination with traditional combination and permutation. Combinatorics is a collection of several computing techniques used in science and management. In this research paper, new results on both combination and permutation are introduced for the...

This paper presents a technique to find the solutions of algebraic equations taken away from the different combinatorics geometric series. This idea can enable the scientific researchers to solve the real life problems