Need help understanding book description for a combinatoris problem

[fleazo]

*combinatorics*, sorry for the typo in the title. The problem is as follows: In how many ways can we write the number 4 as the sum of 5 non-negative integers?

I have taken a screen cap of the solution that my book provides. Here it is

http://imgur.com/8BhxXPq

So I understand the concept of how they're explaining this - for example - one possible selection is I chose "box 1" 2 times, "box 2" 1 time, "box 3" 0 times, "box 4" 2 times, and "box 5" 0 times, this corresponds to : 2 + 1 + 2 = 5.

The confusion for me is, it seems like this approach would have a lot of repeats. For example - the selection of "box 1" 0 times, "box 2" 0 times, "box 3" 2 times, "box 4" 1 time, and "box 5" 2 times, would also produce : 2 + 1 + 2 = 5.I realize I'm missing something in the logic, but I'm not sure what.

 

[fleazo]

oops... I realize now this is a textbook problem and not allowed here. I'm moving it to the textbook problem section. Sorry about that.