Find the coefficient of $x^{2009}$ in the expansion of ${(1 - x)}^{2008} {(1 + x + x^{2})}^{2007}$

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Solution

$f (x) = {(1 - x)}^{2008} \times {(1 + x + x^{2})}^{2007} f (x) = {(1 - x)}^{1} \times {((1 - x) (1 + x + x^{2}))}^{2007} f (x) = {(1 - x)}^{1} \times {(1 + x + x^{2} - x - x^{2} - x^{3})}^{2007} f (x) = {(1 - x)}^{1} \times {(1 - x^{3})}^{2007} f (x) = {(1 - x^{3})}^{2007} - x {(1 - x^{3})}^{2007} A l l t h e t e r m s i n t h e e x p a n s i o n {(1 - x^{3})}^{2007} i s i n t h e f o r m x^{3 r} A n d i n t h e - x {(1 - x^{3})}^{2007} i s o f t h e f o r m x^{3 p + 1} S o 3 r = 2009 o r 3 p + 1 = 2009 W h e r e w e d o n o t g e t i n t e g e r v a l u e s o f r, p . S o t h e c o e f f i c i e n t o f x^{2009} i s 0$

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