Question

In the expansion of ${\left(1+x+x^3+x^4\right)}^{10}$, the coefficient of $x^4$ is

• A. ${}^{40}{C}_4$
• B. ${}^{10}{C}_4$
• C. $210$
• D. $310$

Answer 1

Answer: (D)

${\left(1+x+x^3+x^4\right)}^{10}={\left[(1+x)\left(1+x^3\right)\right]}^{10}$
$=(1+x{)}^{10}{\left(1+x^3\right)}^{10}$
$=\left(1+{}^{10}{C}_{1}x+{}^{10}{C}_2{x}^2+{}^{10}{C}_3{x}^3+{}^{10}{C}_4{x}^4\dots \right)$
$\times \left(1+{}^{10}{C}_1{x}^3+{}^{10}{C}_2{x}^6\dots \right)$
$\therefore$ Coefficient of $x^4=\left({}^{10}{C}_1\right)\left({}^{10}{C}_1\right)+{}^{10}{C}_4$
$=100+\frac{10\cdot 9\cdot 8\cdot 7}{1\cdot 2\cdot 3\cdot 4}$
$=100+210=310$