This paper presents the summations of binomial expansions with the optimized binomial coefficients and also the sum of multiple summations of the binomial coefficients. This combinatorial result is useful for researchers who are working in computing, science, engineering, and management

This paper focuses on the successive partition method applied to a binomial coefficient in combinatorial geometric series. The coefficient for each term in combinatorial geometric series refers to a binomial coefficient. These ideas can enable the scientific researchers to solve the real life...

This paper presents the binomial expansion and series with optimized binomial coefficients. The binomial series is built on the building blocks of optimized binomial coefficients and multiple summations of geometric series