Question

If $|x|<1,$ then the coefficient of ${{x}^{6}}$ in the expansion of ${{(1+x+{{x}^{2}})}^{-3}}$ is

• A. 3
• B. 6
• C. 9
• D. 12

Answer 1

Answer: (A)

3

Answer 2

${{(1+x+{{x}^{2}})}^{-3}}={{\left[ \frac{1}{(1+x+{{x}^{2}})} \right]}^{3}}$
$={{\left[ \frac{1-x}{1-{{x}^{3}}} \right]}^{3}}$
$={{(1-x)}^{3}}{{(1-{{x}^{3}})}^{-3}}$
$=(1-{{x}^{3}}-3{{x}^{2}}+3x)(1+3{{x}^{3}}+6{{x}^{6}}+.....)$
$\therefore$ Coefficient of ${{x}^{6}}$ in ${{(1+x+{{x}^{2}})}^{-3}}$
$=6-3=3$