On the number of words of $N=3 \,n$ letters with a three-letter alphabet

TL;DR

This paper investigates the enumeration of words of length 3n over a three-letter alphabet, categorizing them based on their length's remainder modulo three, and deriving related sums of trinomial coefficients.

Contribution

It provides a detailed counting method for specific word lengths and connects the problem to sums of trinomial coefficients, offering new combinatorial insights.

Findings

Derived formulas for counting words of length 3n with specific properties

Connected word counting to sums of trinomial coefficients

Classified words based on their length modulo three

Abstract

In this paper we address the well-known problem of counting the number of $3n$-letter words that can be formed from a three-letter alphabet by decomposing it into four possible cases based on its remainder when divided by three. The solution to the problem also gives us some sums of trinomial coefficients.