Coefficient of  $x^4$in ${\left(1+x+x^2+x^3\right)}^{11}$is-

 

We have 

$\begin{array}{l}1+x+x^2+x^3=1+x+x^2(1+x)=(1+x)\left(1+x^2\right)\\ \Rightarrow {\left(1+x+x^2+x^3\right)}^{11}=(1+x{)}^{11}{\left(1+x^2\right)}^{11}\\ =\left(1{+}^{11}{C}_{1}x{+}^{11}{C}_2{x}^2{+}^{11}{C}_3{x}^3+\right.\\ \left.{ }^{11}{C}_4{x}^4+\cdots \right)\left(1{+}^{11}{C}_1{x}^2+\right.\\ \left.{ }^{11}{C}_2{x}^4+\cdots \right)\end{array}$

Thus, coefficient of $x^4$in  ${\left(1+x+x^2+x^3\right)}^{11}$

$\begin{array}{l}=\left(1\right)\left({ }^{11}{C}_2\right)+\left({ }^{11}{C}_2\right)\left({ }^{11}{C}_1\right)+\left({ }^{11}{C}_4\right)\left(1\right)\\ =55+\left(55\right)\left(11\right)+330=990\end{array}$