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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.

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1 Answer 1

up vote 2 down vote accepted

Don't worry, the only problem is a simple typo. For 2,2,17, the coefficient should be $$\frac{21!*2^2}{2!*2!*17!}.$$ Instead you imputed $$\frac{21!*2^2}{2!*2!*17}$$ into Wolfram Alpha. Notice there is a 17 in the denominator instead of $17!$. Once this is corrected, the answers do indeed match.

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Oh boy. Thank you this was driving me nuts! –  iDontKnowBetter Sep 17 '11 at 20:53

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