This monograph is the second in a series of such works addressed to the advanced high school student who has an interest in being exposed to mathematical ideas and concepts that are not a part of a standard high school mathematics curriculum. The intent is to
expose the student to the enormous breadth and beauty of mathematics, much of which is lost to a curriculum narrowly focused on preparing students for standardized tests. The hope is that this exposure will excite and encourage the student to view the discipline with an enhanced appreciation.

The first monograph, A Finite Introduction To Infinity, introduced the reader to transfinite numbers: an infinity of infinities. Using simple mathematical concepts, the monograph guided the user through distinctions such as countable versus uncountable infinities and related such magnitudes to various sets, such as the sets of Integers, Rationals, and Real numbers and an extraordinary construct referred to as Cantor Dust. It then launched into its culmination: The Banach-Tarksi Paradox. It was a wild, and hopefully, enjoyable exposition into an infinitely fascinating topic [pun intended].

In this volume we investigate various way to count. Yes...count.
Please resist the urge to stop reading right here and now! Be open to the possibility that, as with Infinity, your preconceived ideas about what counting means may be incomplete. For example, in the first monograph, A Finite Introduction To Infinity we discovered that adding new elements to an infinite set does not necessarily increase its size! This notion of counting required the explicit establishment of a bijection in order to quantify the 'count' (e.g., cardinality) of a set and to compare cardinalities across sets.

For some applications, counting requires one to distinguishing between differing characteristics among the elements under consideration. For example, consider a drawer full of socks of different colors where the socks are randomly distributed. To ask questions such as 'What is the probability of randomly picking 2 matching socks with just two attempts' or 'How many different ways are there to randomly select 3 red socks in 5 attempts? ' involves 'counting', but it must be carried out with special care. As another example, consider the card game Poker, in particular 5-card poker. To determine the odds of being dealt a particular poker-hand, say 3 of a kind, one must count the number of ways to construct such a hand and divide that number by the number of all possible hands (which requires its own count).

There are special techniques for 'counting' in these and other like situations. In this monograph, we discuss and investigate a number of techniques that facilitate the ability to count in the context of complex scenarios, perhaps constrained by certain assumptions.

Through the selection of interesting problem domains, the reader will be introduced to advanced mathematical topics generally not encountered outside of a college level mathematics program; such as Stochastic Processes and Queueing Theory. Although the conceptual
ideas will be new and challenging, the mathematical machinery supporting these concepts should be familiar and accessible to the interested high school student.
About the Author: Rich Coco is a retired "Mathematicaphile", currently residing in the
North Shore of Massachusetts.

After graduating from Woburn High School in May 1970, Rich attended UMass-Amherst that Fall. It was here at UMass that Rich fell hard for both Mathematics and his eventual wife Ida (as fate would have it, in that order). After graduating with a BA-Math degree, Rich was accepted into the Doctorate program (UMass-Amherst) for Mathematics in 1975.

In the summer of 1978, Rich fled the Doctorate program (with a Master's Degree) for a seductive job opportunity at AT&T Bell Laboratories. He always imagined he would continue to pursue his Mathematical interests and finalize his PhD while working. Alas, life, constantly evolving technical direction at work, and an eventual transition into Software Engineering torpedoed any such fantasy. The final nail in that coffin was an in-house 2-year Masters Degree equivalency program in Software Engineering at ATT-BL.

As he matured as a Software Engineer, Rich eventually left AT&T and worked at numerous startup companies and Enterprises (eg, Cisco, EMC, Schneider-Electric) as a Principal Engineer or Software Architect and had the good fortune to work across many different technology domains: Network Management, Crypto Security, Video Conferencing, and Tele-Medicine; to name a few.

Through it all, Rich never quite got over his fascination with Mathematics. Now retired, he finally gets to re-engage with it. The intention is to write a series of monographs on special topics for the enthusiastic High School student, introducing him/her to topics, concepts, and ideas that they will not encounter in a standard High School curriculum. Though such topics and fields of endeavor are generally not encountered until one studies Mathematics formally
at the college level, Rich believes many of them can be made accessible and comprehensible to an interested and motivated High School student. In this way, the student will get a glimpse into the extraordinary breadth, depth, and beauty of the field of study referred to as Mathematics. This will be a perspective that is inaccessible in the standard test-driven High School Mathematics curriculum.