Calculation of Binomial and Multinomial Coefficients by Sequences of Summations

Binomials and multinomies are mathematical functions that do appear in many fields like linear algebra, calculus, statistics and probability, among others. Complete binomial and multinomial construction can be a hard task; there exist some mathematical formulas that can be deployed to calculate binomial and multinomial coefficients, in order to make it quicker. The purpose of this document is the development of an alternative method in order to carry out the calculation of binomial and multinomial coefficient; here are raised three analytic formulas that yield those coefficients for each term by means of summations series. Throughout this document firstly it is exposed the deduction of the two formulas to calculate binomial coefficients, afterwards this result is extended alongside the binomial theorem for the n terms of a multinomial to code a formula that can be used for multinomies. Finally an exemplary case of study is bestowed in order to illustrate how to deploy those formulas.