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

Let, $y={(1-x)}^{2008}{(1+x+{x}^{2})}^{2007}$y=(1−x)2008(1+x+x2)2007

$$\Rightarrow y=(1-x){(1-x)}^{2007}{(1+x+{x}^{2})}^{2007}$$⇒y=(1−x)(1−x)2007(1+x+x2)2007

$$\Rightarrow y=(1-x){[(1-x)(1+x+{x}^{2})]}^{2007}$$⇒y=(1−x)[(1−x)(1+x+x2)]2007

$$\Rightarrow y=(1-x){(1-{x}^{3})}^{2007}$$⇒y=(1−x)(1−x3)2007

$$\Rightarrow y=(1-x)(C02007-C12007({x}^{3})+......)$$⇒y=(1−x)(C02007−C12007(x3)+......)

General term in the expansion is given by,

$${T}_{r+1}=(1-x)(Cr2007{x}^{3r}{(-1)}^{r})$$Tr+1​=(1−x)(Cr2007x3r(−1)r)

$$\Rightarrow {T}_{r+1}{=}^{2007}{C}_{r}{x}^{3r}{(-1)}^{r}-{}^{2007}{C}_{r}{x}^{3r+1}{(-1)}^{r}$$⇒Tr+1​=2007Cr​x3r(−1)r−2007Cr​x3r+1(−1)r

For coefficient of ${x}^{12}$x12,

$$3r=2012$$3r=2012

But, $r\neq \frac{2012}{3}$r=32012​

Also, $3r+1=2012$3r+1=2012

$$\Rightarrow 3r=2011$$⇒3r=2011

But, $r\neq \frac{2011}{3}$r=32011​

Hence, there is no term containing ${x}^{2012}$x2012.

So coefficient of ${x}^{2012}=0$x2012=0