Dear
Gnpth,
I'm happy to respond.
First of all, here is a blog article: it may not be immediate obvious why it's relevant to your question, but it is incredibly pertinent:
https://magoosh.com/gmat/2013/how-to-do- ... th-faster/What's most relevant about that blog is the distinction of Left Brain vs. Right Brain. The Left hemisphere of the brain is all about logic, organization, and precision. It loves clear and unambiguous rules & recipes to follow. You can get through pretty much all of arithmetic & algebra on the left brain alone --- if you just follow the precise rules, you will get the answer right almost every time.
Combinations & probability are very different. They are about the right brain. The right hemisphere of the brain is the side that deals with pattern-matching, poetry, images, symbols, analogies, metaphors, etc. This side of the brain is actually much much faster, but it's not about clear precise rules. Combinations & probability are topics that routinely frustrate folks who try to do all of math in a left-brain mode. You see, there are some rules, but they are not automatic recipes for solutions, as the rules in algebra are.
Here's what you need to do. First of all, you need to learn more about how to access and use your right hemisphere. Some suggestions are given in that article.
With combination & probability problems, you need to read and re-read the solutions to every problem you got wrong. You probably already have read them, but I am going to suggest that you probably didn't read them the right way. You were probably looking for "what should I do?" That's a left-brain question: what should I do? The right-brain question is: how should I look at the problem? how should I conceptually frame the information? That's what you need to discern in each of these problems. It's not so much about the steps you take --- once you frame the problem correctly, the steps to take are obvious. It's all about that initial framing of the information, the initial perceptual choices made. THAT is what you need to get from the solutions. As much as is possible, you need to articulate that: force yourself to write down how the author of the problem explanation began the problem, how that author initially looked at the problem. Forcing yourself to articulate this in words will help you build connections between the two hemispheres of your brain. You have to start thinking, not only about
what to do, but more importantly in these problem,
how to see, that is, the perceptual choices you have available.
I realize, this may be very frustrating advice. You may just want a set of clear rules. You may be thinking, "Just tell me what to do in these problems." Those are all from the left-brain perspective, and by themselves, they do not give you access to the right-brain. There's no left-brain way to get into the perspective of the right-brain. Gaining access to the right-brain perspective is like opening an entirely new perceptual world.
Does all this make sense?
Mike