I started thinking about the idea that a normal curve can be found if you flip a large set of coins many times and count how many heads you get each time. Then plot how many times you got each possible number of heads. You'll end up getting no heads or all heads on all the coins very few times and about half heads the most. This makes a normal curve.
Well I started thinking what if you take the randomness out, and assume that you flipped every possible outcome of a series of heads and tails for say, 10 coins, exactly once. There would be 2^10 or 1024 possible outcomes, each one would show somewhere between 0 total heads or all 10 heads. I also figured that one can use the combinatoric formula called n choose r for telling how many ways there were to get each total number of heads. For instance, for 10 coins and you want to know how many ways there are to have 5 heads on the coins you can do 10 choose 5 or 10!/5!*(10-5)! = 252. Maple the math program actually took a few seconds and derived this formula:
So the next thing was to graph the combinatoric part over the 2^10 whole as you vary r from 0 to 10:
Maple has a way of estimating factorials even in between integers where factorial isn't typically thought of existing but as you can see this leads to a nice normal looking curve. The height is different than the standard normal distribution, though, here is the formula and graph one typically uses:
You can see this formula has all kinds of e's and pi's whereas mine has only factorials and 2^n. But the overall shape is very similar.