Question

The coefficient of $x^4$ in the expansion of ${\left(1-x-2x^2\right)}^8$ is

• A. $144$
• B. $-144$
• C. $154$
• D. $-154$

Answer 1

Answer: (D)

${\left(1-x\left(1+2x\right)\right)}^8$
$=1{-}^8{C}_{1}x\left(1+2x\right){+}^8{C}_2{x}^2{\left(1+2x\right)}^2$
${-}^8{C}_3{x}^3{\left(1+2x\right)}^3{+}^8{C}_4{x}^4{\left(1+2x\right)}^4.........$
The coefficient of $x^4$ is
${}^8{C}_2\cdot 2^2{-}^8{C}_3\cdot 2\cdot 3{+}^8{C}_4$
$=\frac{8\times 7}{2}\times 2^2-\frac{8\times 7\times 6}{3\times 2}.3.2+\frac{8\times 7\times 6\times 5}{4\times 3\times 2}$
$=112-336+70$
$=-154$