$\text{ The coefficient of }{x}^{10}\text{ in the expression }{\left(1+x+x^2+x^3\right)}^5$

${\left(1+x+x^2+x^3\right)}^5={\left((1+x)+x^2(1+x)\right)}^5$

$\begin{array}{l}={\left((1+x)\left(1+x^2\right)\right)}^5\\ =(1+x{)}^5{\left(1+x^2\right)}^5\end{array}$

$\begin{array}{l}=\left(5_{C_0}+5_{C_1}x+5_{C_2}{x}^2+5_{C_3}{x}^3+5_{C_4}{x}^4+5_{C_5}{x}^5\right)\times \\ \left(5_{C_0}+5_{C_1}{x}^2+5_{C_2}{\left(x^2\right)}^2+5_{C_3}{\left(x^2\right)}^3+5_{C_4}{\left(x^2\right)}^4+5_{C_5}{\left(x^2\right)}^5\right)\end{array}$

$\text{∴Coefficient of }{x}^{10}\text{ is }$

$\begin{array}{l}=\left({}^{5}C_0\times {}^{5}C_5+{}^{5}C_2\times {}^{5}C_4+{}^{5}C_4\times {}^{5}C_3\right)\\ =1\times 1+10\times 5+5\times 10\\ =1+50+50\\ =101\end{array}$