Counting With Intent

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.