Mathematics - Binomial Theorem Question with Solution | TestHub

${\left(1+x+x^3+x^4\right)}^{10}={\left[\left(1+x\right)\left(1+x^3\right)\right]}^{10}$

$={\left(1+x\right)}^{10}{\left(1+x^3\right)}^{10}$

$=\left(1+{}^{10}C_{1}x+{}^{10}C_2{x}^2+{}^{10}C_3{x}^3+{}^{10}C_4{x}^4\dots \right)\times \left(1+{}^{10}C_1{x}^3+{}^{10}C_2{x}^6+\dots .\right)$

$\therefore$ Coefficient of $x^4=\left({}^{10}C_1\right)\left({}^{10}C_1\right)+{}^{10}C_4$

$=100+210=310$