The coefficient of $x ^ { 4 }$ in the expansion of $\left( 1 + x + x ^ { 2 } + x ^ { 3 } \right) ^ { 11 }$ is

$\begin{array}{l} \left(1+x+x^{2}+x^{3}\right)^{11} \\ =\left(1+x^{2}\right)^{11}(1+x)^{11} \\ =\left(1+11 x^{2}+{ }^{11} C_{2} x^{4}+\ldots x^{22}\right)\left(1+11 x+{ }^{11} C_{2} x^{2}+\ldots x^{11}\right) \end{array}$

Hence coefficient of $x^{4}$ will be

$\begin{array}{l} 1 \times{ }^{11} \mathrm{C}_{4}+11 \times{ }^{11} \mathrm{C}_{2}+{ }^{11} \mathrm{C}_{2} \times 1 \\ ={ }^{11} \mathrm{C}_{4}+12 \times{ }^{11} \mathrm{C}_{2} \\ =330+660 \\ =990 \end{array}$