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Question 82

The coefficient of $$x^{2012}$$ in the expansion of $$(1 - x)^{2008}(1 + x + x^2)^{2007}$$ is equal to _____.


Correct Answer: 0

$$(1-x)^{2008}(1+x+x^2)^{2007}$$

Note: $$1 + x + x^2 = \frac{1-x^3}{1-x}$$ (for $$x \neq 1$$).

$$(1-x)^{2008} \cdot \left(\frac{1-x^3}{1-x}\right)^{2007} = (1-x)^{2008-2007} \cdot (1-x^3)^{2007} = (1-x)(1-x^3)^{2007}$$

We need the coefficient of $$x^{2012}$$ in $$(1-x)(1-x^3)^{2007}$$.

$$= \text{coeff of } x^{2012} \text{ in } (1-x^3)^{2007} - \text{coeff of } x^{2011} \text{ in } (1-x^3)^{2007}$$

Coeff of $$x^{3k}$$ in $$(1-x^3)^{2007}$$ is $$(-1)^k\binom{2007}{k}$$.

$$x^{2012}$$: $$2012/3$$ is not an integer, so coefficient = 0.

$$x^{2011}$$: $$2011/3$$ is not an integer, so coefficient = 0.

Answer: $$0 - 0 = 0$$.

The answer is $$\boxed{0}$$.

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