Question

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

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

Answer 1

Answer: (D)

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