I cant understand probability and combinatorics, even if I think that I understand the fundamentals............

I have to smile at this question.

For a long time, I considered myself in the same boat. Hell, I still consider myself in the same boat, even if I know a lot more probability and combinatorics than I used to. My knowledge in this area has always been grossly disproportionately small when compared to my knowledge in any other area of mathematics.

If you’re still early in your mathematical journey, perhaps I can help a little. I have a two-part suggestion:

First, slow down and figure things out by first principles. Use formulas and algebraic manipulations only as a last resort.

In other words, if I asked you how many ways there are to make a committee of 5 people from a class of 30 people, you shouldn’t think 30!/(25!⋅5!)$30!/(25!\cdot 5!)$. You should think (305)$(\genfrac{}{}{0ex}{}{30}{5})$. And maybe even to take even a further step back, you shouldn’t instantly think (305)$(\genfrac{}{}{0ex}{}{30}{5})$, but instead kind of think about what that expression means, and whether it really is the right tool for the job.

Another fundamental question that sometimes trips people up is when to add and when to multiply. I won’t attempt to explain it here (even though it’s straight forward). Instead, I just want you to sensitize you to that question.

Then do a bunch of problems. You’ll be really slow. Go back and check your work, and hopefully learn from the solutions. Flag those problems as problems you missed, and do them again in a few days after you forgot at least a little bit about the solution.

Where to get the problems? This might be a little old fashioned, but:

I’m a big fan of this series of books. I don’t know how old they are, but I loved them in the early/mid-90s. I doubt they have been updated much, except maybe for the typography.

But they’re really great books, because they have a ton of problems and solutions that are “parsed” very well. What I mean is, you’ll see at least one problem that features every line of thought (or outright trick) you’ll need to know. And many more that you don’t.

If you hunker down and do every problem in this book, you’ll be a master. If you hunker down and do half the problems in this book, you’ll be an expert.

-Charles Slade