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.