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Whenever were doing combinations or permutations, aren’t we always using the product rule? So i can write out the phases for every counting problem and then i determine whether i overcount or not. Usually if its choosing things of the same type, i see that i end up overcounting by the ways you can permute it but if its choosing things that are diff types, like putting caps on different color markers, i am not overcounting because the order in which you choose a cap per colored marker matters. If this is totally incorrect, which it probably is, how does one know when to overcount or not?

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  • $\begingroup$ It seems to me that "counting problems" include more than just what you can solve with combinations or permutations. Presumably you've seen a few examples of counting problems. If you don't change the problem too much, your generalization might apply. Otherwise it wouldn't. $\endgroup$ – David K Mar 11 at 3:52
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    $\begingroup$ In particular, the Principle of Inclusion-Exclusion is waiting to blow your mind. $\endgroup$ – Gerry Myerson Mar 11 at 3:57
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    $\begingroup$ One relatively general approach to counting is the Twelvefold Way. See users.math.msu.edu/users/magyar/Math482/Twelvefold-Way-482.pdf for example. But also see proctor.math.unc.edu/30fold.html, claiming another 18 types of counting. And I guess there are counting problems that are not included even in that classification. $\endgroup$ – David K Mar 11 at 4:03

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