Repetition in Colored Sequences of Balls

TL;DR

This paper investigates the combinatorial problem of counting colored sequences with repeated colors, providing formal formulations and solutions for various problem variations.

Contribution

It introduces a formal framework for counting sequences of colored balls with repetitions and solves multiple variations of the problem.

Findings

Derived formulas for counting sequences with repeated colors.

Solved several variations of the problem.

Provided a formal mathematical framework.

Abstract

In responding to a question on Math Stackexchange, the author formulated the problem of determining the number of strings of balls colored in most $n$ colors with a number $k$ of repeated colors. In this paper, we formulate the problem more formally, and solve several variations.