Thank you! I think I have it now. This helped a lot. I ended up thinking about it in terms of boxes -- if I have n choose r (what this means is undefined for n

Thank you! I think I have it now. This helped a lot. I ended up thinking about it in terms of boxes -- if I have n choose r (what this means is undefined for now) boxes that all contain r things chosen from n, and choose one box, I now have to choose an ordering for the items in the box. There are r! options for this step. This procedure gives the number of r-permutations of n objects, which is equal to n!/(n-r)! Dividing through by r! gives the desired result. It also gives me an understanding of why dividing by r! on the right still gives an integer result