Mathematics - Binomial Theorem Question with Solution | TestHub

$\left(1+x\right){\left(1-x\right)}^{10}{\left(1+x+x^2\right)}^9$
$=\left(1+x\right)\left(1-x\right){\left(1-x\right)}^9{\left(1+x+x^2\right)}^9$
$=\left(1-x^2\right){\left[\left(1-x\right)\left(1+x+x^2\right)\right]}^9$
$=\left(1-x^2\right){\left(1-x^3\right)}^9$
So, Coefficient of$x^{18}$in$\left(1+x\right){\left(1-x\right)}^{10}{\left(1+x+x^2\right)}^9$
$=$Coefficient of$x^{18}$in$\left(1-x^2\right){\left(1-x^3\right)}^9$
$=$Coefficient of$x^{18}$in$\left(1-x^2\right)\left[{}_{ }{}^9{C}_0+{}_{ }{}^9{C}_1\left(-x^3\right)+{}_{ }{}^9{C}_2{\left(-x^3\right)}^2+\dots \right]$
$={}_{ }{}^9{C}_6{\left(-1\right)}^6={}_{ }{}^9{C}_3=\frac{9.8.7}{1.2.3}=84$