It's definitely not a language problem, I'm a native English speaker...I guess I meant not "why do you divide?" in the sense that the answer is "to remove over counting" (which I clearly knew since I said that in my original post), I meant "why does dividing work?" But either way, I didn't say "Yes, I know the *reason*, but why?" I said "Yes, I know the *reason*, but how was that conclusion arrived at?"

and as for what I mean by a formal proof, I mean something like

this proof of the binomial theorem formula