Originally Posted by 22B
For comparison, the K12Inc Prealgebra course simply stated formulas (after adequately defining n!)
P(n,r)=n!/(n-r)!
C(n,r)=n!/[(n-r)!r!]
without any explanation of why LHS=RHS. (Usually they give some kind of explanation for things, but not this time.) Then they have some "worked examples" which are easily skipped past, then a routine quiz.

How does AoPS take a student who hadn't seen n! before, and lead them to know these formulas and understand why they are true? And what other activities surround the learning of that particular piece of maths? (Or substitute any other piece of maths for the purposes of this discussion.)

How does AoPS work?

I agree with others... the K12 approach is awful.

Have you seen the AoPS promotional mega-sample at http://www.artofproblemsolving.com/Resources/Files/Excerpts.pdf

Chapter 4 in that file is on divisor counting. The introductory material is basically a special case of regular permutations. Looking through that should give you an idea of how they would approach a somewhat more advanced combinatorics topic.

I didn't see any similar topics in their preA book on a quick glance through. The chapter above is from the Number Theory book which is after Algebra.

The sample has lots of good material to get a feel for how they teach. It might be more useful to you than the shorter book samples. I have no experience with the classes.