Award Date


Degree Type


Degree Name

Master of Science (MS)


Mathematical Sciences

First Committee Member

Peter Shiue

Second Committee Member

Gennady Bachman

Third Committee Member

Michelle Robinette

Fourth Committee Member

Laxmi Gewali

Number of Pages



The study of this paper is based on current generalizations of Pascal's Triangle, both the expansion of the polynomial of one variable and the multivariate case. Our goal is to establish relationships between these generalizations, and to use the properties of the generalizations to create a new type of generalization for the multivariate case that can be represented in the third dimension.

In the first part of this paper we look at Pascal's original Triangle with properties and classical applications. We then look at contemporary extensions of the triangle to coefficient arrays for polynomials of two forms. The first of a general polynomial in one variable with terms of each power and coefficients of one, the second the sum of an arbitrary number of terms typical to a multinomial expansion.

We look at construction of the resulting objects, properties and applications. We then relate the two objects together through substitution and observe a general process in which to do so.

In the second part of the paper I observe an application of the current generalizations to the classical problem "The Gambler's Game of Points" to games of alternative point structures. The paper culminates with a generalization I have made for a particular case of the second equation, moving the current four dimensional generalization into the third dimension for observation and study. We see the relationships of this generalization to those from our overview in part one, and develop the main theorem of study from the construction of its arrangement. From this theorem we are able to derive several interesting combinatorial identities from our construction.


Combinatorial analysis; Combinatorics; Pascal's triangle


Discrete Mathematics and Combinatorics | Mathematics