I'm having substantial difficulty learning counting. The subject appears to be presented as a large, scattered collection of contrived formulas, few of which are obvious, with very little explanation of the relationships between them. Ideally, there would be a master formula from which all specific formulas could be derived, but this may be an unrealistic expectation.
The next best thing would be to learn a list of independently toggleable feature distinctions, sufficiently jointly exhaustive to narrow a word problem down to a unique formula. Examples of such feature distinctions include "arrangement order matters VS arrangement order doesn't matter," "elements are uniquely identifiable (each red ball has a serial number) VS elements are not uniquely identifiable (a red ball is a red ball)," and "replacement VS no replacement." Then a set of tables (or a hierarchical database?) would be supplied to reference the formula for each combination of features, all in the framework of a single, consistent notation. The learner could use these tables to solve problems, and memorize the formulas over time, similar to a Laplace Transform table in an ODE course.
I started to create my own set of tables here for the $3$ distinctions above, but it needs to be expanded to consider the features that distinguish arrangements, derangements, ordered and unordered set partitions, number theory partitions, compositions, etc., as well as any distinctions within those topics. This is a very tall task for someone learning the subject, and would be more appropriate for an educator to organize. I've been finding it prohibitively time consuming and difficult to piece together loads of scattered formulas, try to parse out the fundamental features distinguishing them, and ask tons of questions about combining those features in the other, untaught ways.
The formulas corresponding to some combinations of features are unknown, of course, but at least the approach described would give a reasonable, systematic way to assess the relevant features of a real world counting problem when the appropriate formula is known. Without this degree of organization, I feel like what I'm doing is the equivalent of trying to learn Laplace Transforms in an ODE class that offers no LT definition, no reference table, and scatters the individual definitions for the LTs of specific functions randomly throughout the course, except that the combinatorics formulas I'm trying to learn are far more numerous.
Have there been any attempts to introduce counting in a more organized, top-down fashion? To summarize, I'm having a hard time both sharply understanding/piecing together the various distinctions present in arrangements, derangements, partitions, etc. (i.e., "order mattering" is used with different meanings in not-entirely-dissimilar contexts), and breaking word problems and real world problems down in terms of these distinctions. I'm certainly not seeking rigor for the sake of rigor, but presentations of this subject I've seen seem to use omission, rather than curation, to increase the accessibility of material, which for me is having the opposite impact.