Given 5 children and 8 adults, how many ways can they be seated so that there are no two children sitting next to each other. ( From math.stackexchange )
I haven't opened the question yet. I got alerted by this question that my discrete mathematics skills are getting -sloppy-! It happened to me before. There was a time when I thought that I had forgotten all of my linear algebra skills. I can assure you that going through all that material again feels overwhelming. Counting is an essential skill I am going through it again. The theory won't be the problem. I know the formulas, it's the skill in which theorems to apply to a certain to problem, or how to model a counting problem. Boxes or balls? Repetition, yes or no? Distinct or similar objects? Should I use the addition or product rule?
As elementary number theory, enumerative combinatorics ( = counting ) is part of the Olympiad curriculum, so there are TONS of practice questions 'out-there' (*).
(*) Anywhere from Amazon to IRC #bookz channels to the shadow-Internet. What suits you (r budget ) best. As long as you are learning.
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