I think I've discovered that over the past week and a half. But after about four days spent on permutations and combinations with my elementary ed class, today they got it.
We've talked about choosing the first item and the second and the third, and so on. We've done examples and written down lists of choices. We've played with reorganizing manipulatives in different ways. We've used formulas and related these to the ideas we've already discussed. And we've done so many problems I've mostly run out of problems to assign from the textbook. (There weren't all that many to begin with, 'though.)
But on Friday, they understood. We checked the homework, we discussed how to lay down groups of different colored markers in a row, and we even discussed how you could count the number of possible ways to get five heads and five tails when you flip ten coins. And they got it. Very few people said "I don't understand." Hardly anyone asked "Why did you divide by ___?" I could have sung when I left the room. I think I did.
The only thing that worries me: As we get ready to move into geometry, do I have four days on perimeter and area to look forward to?
Of course, this was all Friday. Today, we did a few more examples, and these were a little weaker. Oh well; on to geometry.