Something deeply unsettling is afoot in the land of math education when I'm teaching the same backwards thing in the same backwards way it was presented to me as a high school kid. To wit, combinations.
Here is the current state of the art, according to the big boys of Advanced Algebra publishing:
Fundamental Counting Principle ====> Permutations ====> Combinations ====> Pascal's Triangle ====> Binomial Theorem ====> Celebration
I submit that, when we do it this way, we're double-charging our students for their attention. We bog them down in unnecessary algebraic trifling, and we go out of our way to delay the payoff for just as long as possible. It's bad marketing, and it's bad teaching. And we don't exactly get away scot-free in all this. I know I'm in for a rough couple of weeks any time I have to close my opening lesson with, "Trust me."