What is a exhaustive book or resource for explaining combinations, probabilities, and permutations with practice?

Question

I know how to do combinations and permutations, sort of, but my intuition is still slow so I figured I just need practice after practice. I can do some questions if it is set up a certain way, but once they change it a bit, I get thrown off.

What are the best books/websites where I can practice endless problems until I can do them with ease? I'm hoping such a book will have countless practice problems and good tips and tricks for doing these problems. For example, if order doesn't matter that means that the number of outcomes should be less so you should divide by an additional factorial, etc...

Additionally, is there a good way to approach each problem? Like a tried and true step by step process? I'm trying to formulate a step by step process I can use, which hopefully will save a lot of time, but not sure if full proof. For example:

• Step 1: Always calculate the total number of possible outcomes. This will be the denominator.
• Step 2: Determine the number of outcomes that will fit the problem's criteria. This will be the numerator.
• Step 3: Divide the two and you get your answer.

Or something like:

• Step 1: Calculate the probability of one specific outcome (that fits your criteria) of occurring.
• Step 2: Calculate the number possible ways your outcome can occur.
• Step 3: Multiply the two and get your answer.

For example, let's say the question is, if you draw 3 cards from a 52 card deck, what is the probability that the 3 cards is 2 spades and 1 non-spade. Using the first approach:

• 52! / (3! * 49!)
• 13! / (2! * 11!) * 39
• 3042 / 22100 = 0.138

Using the second approach:

• (13/52) * (12/51) * (39/50)
• 3! / (2! * 1!)
• 0.045882 * 3 = 0.138

I'm having a hard time determine a fast, efficient, and full proof way to approach problems. Or if there is no full proof step by step approach, maybe a high likelihood step by step approach, and then if that doesn't work, use an alternative step by step approach.

Thanks!

Answer 1

I have used Walk Through Combinatorics by Miklos Bona. It covers the things you are interested in, and it has some other interesting applications.

Answer 2

You might consider Schaum's Theory and Problems of Combinatorics, Including Concepts of Graph Theory by V.K. Balakrishhnan. From the front cover: "Includes 600 solved problems with complete solutions and 200 additional problems with answers". What's more, it's relatively inexpensive compared with other books.

That said, and I don't mean to discourage you, I don't think a universal method of solution exists. One of the characteristics of combinatorics is that you can take a fairly easy problem, change it just a little, and all of a sudden you have a much harder problem. It's a bit like number theory in that regard. On the other hand, it definitely helps to work a lot of problems.