Intermediate Counting and Probability (the essential parts)

290 Pages·2007·72.146 MB·English

by  David Patrick

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Printed in 2012.
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HOW TO USE THIS BOOK
Sidenote: This box will contain material which, although interesting, is not part of
t. ,N the main material of the text. It's OK to skip over these báxes, buiif you
read them, you might learn something interestingl
Bogusf Sfoilution: Just like the impossible cube shown to the left, there's something
wrong with any "solution" that appears in this box.
Exercises, Review Problems, and Challenge Problems
Most sections end with several Exercises. These will test your understanding of the material that was
covered in the section. You should try to solve all of the exercises. Exercises marked with a * are more
difficult.
Most chapters conclude with several Review Problems. These are problems that test your basic
understanding of the material covered in the chapter. Your goal should be to solve most or all of the
Review Problems for every chapter-if you're unable to do this, it means that you haven't yet mastered
the material, and you should probably go back and read the chapter again.
All of the chapters (except for Chapter 1) end with Challenge Problems. These problems are
generally more difficult than the other problems in the book, and will really test your mastery of the
material. Some of them are very, very hard-the hardest ones are marked with a *. Don't expect to
be able to solve all of the Challenge Problems on your first try-these are difficult problems even for
experienced problem solvers. If you are able to solve a large number of Challenge Problems, then
congratulations, you are on your way to becoming an expert problem solver!
Hints
Many problems come with one or more hints. You can look up the hints in the Hints section in the
back of the book. The hints are numbered in random order, so that when you're looking up a hint to a
problem you don't accidentally glance at the hint to the next problem at the same time.
It is very important that you first try to solve each problem without resorting to the hints. Only after
vou'r'e seriously thought about a problem and are stuck should you seek a hint. Also, for problems
rrhich have multiple hints, use the hints one at a time; don't go to the second hint until you've thought
about the first one.
Solutions
I:,e solutions to all of the Exercises, Review Problems, and Challenge Problems are in the separate
:-'-:tions book. If you are using this textbook in a regular school class, then your teacher may decide
: --: t,-, make this solutionsbook available to you, and instead present the solutions him,/herse1f. However,
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HOW TO USE THIS BOOK
A Note to Teachers
We believe students leam best when they are challenged with hard problems that at first they may not
know how to do. This is the motivating philosophy behind this book.
Rather tha¡ fust introducing new material and then giving students exercises, we present problems
at the start of each section that students should try to solve before new material is presented. The goal
is to get students to discover the new material on their own. Often, complicated problems are broken
into smaller parts, so that students can discover new techniques one piece at a time. After the problems,
new material is formally presented in the text, and fuIl solutions to each problem are explained, along
with problem-solving strategies.
We hope that teachers will find that their stronger students will discover most of the material in this
book on their own by working through the problems. Other students may leam better from a more
traditional approach of first seeing the new material, then working the problems. Teachers have the
flexibility to use either approach when teaching ftom this book.
The book is linear in coverage. Generally, students and teachers should progress straight through
the book in order, without skipping chapters. Sections marked with a * contain supplementary material
that may be safely skipped. In general, chapters are not equal in length, so different chapters may take
different amounts of classroom time.
Links
The Art of Problem Solving website has a page containing links to websites with content relating to
material i¡r this book, as well as an errata list for the book. This page can be found at:
http artofproblemsolving com/Booklinks/Intermcounting/links php
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ACKNOWLEDGEMENTS
against, arrd socialize with one another, forming friendships and sharpening their mathematical
skitls. The contest is written for high school students, although some exceptional junior high
students attend each year. The competition consists of several events, which include a team
round, a power question (in which a team solves proof-oriented questions), an individual round,
and two relay rounds. More information ls available at ¡.nru. arml . com.
¡ The Harvard-MlT Mathematics Tournament, which is an annual math tournament for high
school students, held at MIT and at Harvard in alternating years. It is run exclusively by MIT and
Harvard students, most of whom themselves participated in math contests in high school. More
information is available at web .mit . edu/hmmt/.
Some other sources for problems in the book are listed below.
MATHCOUNTS MATHCOUNTS, the premier national math contest for U.S. middle school students.
Putnam The William Lowell Putnam Competition, an annual math competition for undergraduate
sLudents in North A meric¿.
PSS Problem-solaing Strategies by Arthur Engel. (See the references on page 366 for the complete
citation.)
ToT The International Mathematics Tournament of Towns, a mathematical problem solving competition
for students throughout the worid.
We have also included problems from various countries' national math Olympiad contests. These
problems are cited by country name in the text.
How We Wrote This Book
This book was written using the tr+TEX document processing system. We thank the authors of the various
trAT¡[ packages that we used while preparing this book, and also the authors of The Wl¿X Companion for
writing a reference book that is not only thorough but also very readable. The diagrams were prepared
using METFPO5T.
About Us
Thls book is a collaboraüve effort of the staff of the Art of Problem Solving. Richard Rusczyk wrote
the original lecture notes for the online course that was the inspiration for this book; he also read
several drafts of this book and made many, many helpful suggestions. Extensive proofreading of the
manuscript was done by Greg Brockman, Chris Chang, Larry Evans, Sean Markan, Jeff Nanney, Adrian
Sanborn, Nathan Savir, and Valentin Vornicu. Special thanks to Hyun Soo Kim for his assistance with
selecting many of the problems. Vanessa Rusczyk designed the cover.
CON.J3NIS
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