I'm trying to find the coefficient of $ x^{36} $ in the expansion of $ (2 - x + x^2)^{21} $
So I found all possible combinations of $ \displaystyle 2^x, x^y, (x^2)^z $ that yield 36 and $ x + y + z = 21 $:
3, 0, 18
2, 2, 17
1, 4, 16
0, 6, 15
Adding those coefficients and plugging those numbers into Wolfram Alpha, I get:
(copy and paste link, because the wolfram link format is getting screwed up)
http://www.wolframalpha.com/input/?i=(%20(21!*2%5E3)%2F(3!*18!))%20%2B%20(%20(21!*2%5E2)%2F(2!*2!*17)%20)%20%2B%20(%20(21!*2)%2F(4!*16!)%20)%20%2B%20(%20(21!%2F(6!*15!)%20)&t=macw01
But this is not even close to the correct coefficient:
http://www.wolframalpha.com/input/?i=(2-x%2B(x%5E2))%5E21&t=macw01
What am I screwing up? I went over those numbers ten times, and they're the only combinations I can think of.