Coefficient of x^(2012) in (1-x)^(2008) (1+x+x^2)^(2007) (1) 0 (2) 1 (3) 2 (4) 3

Question

Coefficient of \(x^{2012}\ \text{in} (1-x)^{2008}\left(1+x+x^2\right)^{2007}\)

(1) 0

(2) 1

(3) 2

(4) 3
 

Answer 1

Correct option is (1) 0  

\((1-x)\left[(1-x)\left(1+x+x^2\right)\right]^{2007}\)   

\(=(1-x)\left(1-x^3\right)^{2007}\)   

\(=\left(1-x^3\right)^{2007}-x\left(1-x^3\right)^{2007}\)   

[\(\left(1-x^3\right)^{2007}\) 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 \(x^{2012}\ \text{in} \left(1-x^3\right)^{2007}=0\)   

Coefficient of \(x^{2011}\ \text{in} \left(1-x^3\right)^{2007}=0\)  

\(\Rightarrow \text{Coefficient of} \ x^{2012}\text{ in} (1-x)^{2008}\left(1+x+x^2\right)^{2007}=0\)