So, this is my problem…I have completed my boards and among all others, I have a great weakness in combinatorics. So this means I can utilize my free time now to address this problem. I think it is because of the fact that my understanding of the basics is not correct.
So, I am asking which book should I consult to have a better understanding of the basics in combinatorics. I would like to have a book which is rigorous and builds up the theory from the beginning(possibly with also a little bit of history). I mainly though aim to have a stronghold on the topics of permutations and combinations, the inclusion-exclusion principle, the pigeonhole principle, colouring proofs(things like covering chessboards) and other things that an interested student ought to know in high school. The book need not have difficult problems, for I can find difficult problems.
One such book(in Number Theory), I have read is Elementary Number Theory by David M. Burton(the first few chapters and bits of later chapters). I telling you about this book so that I can possibly be clear of what kind of book I am looking for.
Best Answer
If you are looking for a high school-level book, you might consider The Mathematics of Choice: Or, How to Count Without Counting by Ivan Niven. I don't think it covers coverings of chessboards, so far as I can recall, but it covers all the other topics in your list. You can buy the book direct from the Mathematical Association of America: http://www.maa.org/publications/maa-reviews/mathematics-of-choice-or-how-to-count-without-counting
This book was my first introduction to combinatorics, way back when I was in high school, and I remember it fondly.