Mathematics - Binomial Theorem Question with Solution | TestHub

We have coefficient of $x^4$ in ${\left(1+x+x^2+x^3\right)}^{11}$ $$\begin{array}{l}=\text{ coefficient of }{\mathrm{x}}^4\text{ in }{\left(1+{\mathrm{x}}^2\right)}^{11}(1+\mathrm{x}{)}^{11}\\ =\text{ coefficient of }{\mathrm{x}}^4\text{ in }(1+\mathrm{x}{)}^{11}+\text{ coefficient }\\ \text{ of }{\mathrm{x}}^2\text{ in }\\ \text{ 11. }(1+\mathrm{x}{)}^{11}+\text{ constant term is }{}^{11}{\mathrm{C}}_2\cdot (1+\mathrm{x}{)}^{11}\\ ={}^{11}{\mathrm{C}}_4+11.{}^{11}{\mathrm{C}}_2+{}^{11}{\mathrm{C}}_2=990\end{array}$$