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

We have, 

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

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

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