Coefficient of \(x^{2012}\) in \((1-x)^{2008}\left(1+x+x^2\right)^{2007}\). (22 Jan, Shift I, Memory Based)

\(
\begin{aligned}
& (1-x)\left[(1-x)\left(1+x+x^2\right)\right]^{2007} \\
& =(1-x)\left(1-x^3\right)^{2007} \\
& =\left(1-x^3\right)^{2007}-x\left(1-x^3\right)^{2007}
\end{aligned}
\)

\([(1- \left.x^3\right)^{2007}\) contains \(3 \lambda\) types of exponents while \(x\left(1-x^3\right)^{2007}\) will have \((3 \lambda+1)\) type while 2012 is
\((3 \lambda+2)\) type] that is not possible \(\Rightarrow 0\)
Coefficient of \(x^{2012}\) in \(\left(1-x^3\right)^{2007}=0\)
Coefficient of \(x^{2011}\) in \(\left(1-x^3\right)^{2007}=0\)
\(\Rightarrow\) Coefficient of \(x^{2012}\) in \((1-x)^{2008}\left(1+x+x^2\right)^{2007}=0\)