Question

The coefficient of $x^6$ in the expansion of ${\left(1+x+x^2\right)}^6$ is

• A. $131$
• B. $141$
• C. $151$
• D. $167$

Answer 1

Answer: (B)

Coefficient of $x^6$ in ${\left(1+x\left(1+x\right)\right)}^6$
$=1+{}^6{C}_{1}x(1+x)+{}^6{C}_2{x}^2(1+x{)}^2+{}^6{C}_3{x}^3(1+x{)}^3+{}^6{C}_4{x}^4(1+x{)}^4$
$\Rightarrow$ The coefficient of $x^6$ is
$\_{}^6{C}_3\times \_{}^3{C}_3+\_{}^6{C}_4\times \_{}^4{C}_2+\_{}^6{C}_5\times \_{}^5{C}_1+\_{}^6{C}_6\times \_{}^6{C}_0$
$=\frac{6\times 5\times 4}{3\times 2}\times 1+\frac{6\times 5}{2}\times \frac{4\times 3}{2}+6\times 5+1\times 1$
$=20+90+30+1=141$