TITLE: What is an intuitive explanation of the combinations formula?
QUESTION [9 upvotes]: I perfectly understand the permutations formula i.e. if you have $n$ things how many ways can you rearrange it if taken $k$ at a time (or if you have $k$ slots)? So you draw the following tree. And the formula comes out naturally.
$$^nP_k = \frac{n!}{(n-k)!}$$

I get that it is including all of the duplicates, too. And to get rid of them we use the combinations formula:
$$^nC_k = \frac{n!}{k!(n-k)!}$$
Can somebody explain why we have to divide by the "number of ways to re-arrange the slots" to get rid of the duplicates? I understand that it works and the division is used to scale a number down but I am not getting that "aha moment" of why we do it. Can somebody explain it?

REPLY [2 votes]: Suppose I have a bag full of $n$ balls, and I want to choose $k$ of them. I pull $k$ balls out and put them to the side, in a nice line. I then tip the balls left in the bag out and arrange them in a nice a line. In total I have $n!$ ways to arrange all of my balls, $k!$ ways to arrange the first line, and $(n-k)!$ ways to arrange the second line. Therefore the number of ways to choose $k$ balls is $\frac{n!}{k!(n-k)!}$: The number of ways I can have $n$ items, divided by the number of ways I can take $k$ and the number of ways I can have $(n-k)$ remaining items arranged.