# Outline for high school combinatorics class?

I am a high school student and I have taken all the math classes that my school provides (through calculus AB). I have been looking at a possible independent study for next year and I have landed on combinatorics and possibly graph theory as well. I plan on using MIT Open courseware's "Combinatorics: The Fine Art of Counting" videos and supplementing them with a textbook.

#### Does anyone have any suggestions for a textbook

1. where the only prerequisite would be AP Calculus AB and would be easily comprehended by a high school student? Ideally the book would delve into graph theory as well (I understand these two subjects go hand-in-hand).

2. to provide solutions (or at least some solutions) to the problems to make sure I am on the right track.

#### 3. Does anyone have an opinion on the following books? I'm worried books from Art of Problem Solving (for middle school/high school students) might be too elementary.

A Walk Through Combinatorics: An Introduction to Enumeration and Graph Theory

Combinatorics and Graph Theory (Undergraduate Texts in Mathematics)

Principles and Techniques in Combinatorics

How to Count: An Introduction to Combinatorics

Combinatorics: A Guided Tour (MAA Textbooks)

Introduction to Counting & Probability by David Patrick

Intermediate Counting & Probability by David Patrick

• I’ve not seen Miklós Bóna’s A Walk Through Combinatorics, but I’ve read good things about it, and I do have a good opinion of his Introduction to Enumerative Combinatorics. You might take a look at Kenneth Bogart’s Combinatorics Through Guided Discovery. Jan 28, 2015 at 0:20
• People generally refer to books by their authors, not their titles. Jan 28, 2015 at 1:33
– user53259
Jul 18, 2021 at 20:33
– user1124753
May 18, 2023 at 6:44

I would consider Mathematics of Choice by Niven and Graphs and Their Uses by Ore. http://www.maa.org/publications/ebooks/anneli-lax-new-mathematical-library

These books are part of a series intended for talented high schoolers, and the authors were first-rate mathematicians.

• While I am not familiar with the book by Ore, the book by Niven is excellent. If I were teaching a course on combinatorics to high school students, Niven's text is the one I would use. Jan 28, 2015 at 12:58

Your question is quite broad, but I agree with Modded Bear that AP calc is unlikely to be of much help. The Concrete Mathematics book would be excellent, but I might throw in the book Discrete Mathematics and Its Applications by Kenneth Rosen. This book is an absolute tome with thousands of exercises (literally) that range from the very easy to the exceedingly difficult. The newest edition of the book is outrageously expensive. Luckily, pretty much all of the resources for the 5th edition of the book may be found online as .pdf files (although I would recommend buying a cheap used edition for the actual textbook):

Having all of the solutions accessible would make this an excellent text to use. Good luck.

• I can substantiate that Kenneth Rosen's book is very good. The 7th edition is very easily located as a .pdf online too. Grimaldi's Discrete and Combinatorial Mathematics covers a good portion of discrete mathematics and the text and solution manual are easily found online in .pdf format. Jan 28, 2015 at 3:16
• @Dunka Yes, Rosen's 7th edition book may easily be found online but the accompanying solutions manuals cannot. If so, please do pass on the links. That's actually why I chose the 5th edition--because I could find everything online. Jan 28, 2015 at 3:17

For combinatorics, I would highly recommend the book by Yao Zhang. Its an excellent wealth of information and provides common strategies for counting that tackle pretty much all problems out there. It has both text that teaches you, example problems, and "challenge" problems. You can find it easily online.

knowing AP calculus will probably not help a lot unless you want to learn generating functions. In my experience you the prerequisites you might not know in some books (although not all) may be linear algebra or group theory.

Here are some books I recommend on the subject:

Concrete Mathematics

Problem solving methods in combinatorics

Bondy and Murty GTWA