Question

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

• A. $990$
• B. $495$
• C. $330$
• D. none of these

Answer 1

The correct option is A $990$

$={(1+x+x^2+x^3)}^{11}$

$={\left(\right(1+x\left)(1+x^2)\right)}^{11}$

$=(1+x{)}^{11}[1{+}^{11}{c}_1{x}^2+{}^{11}{c}_2{x}^4+{\cdots }^{11}{c}_{11}{\left(x^2\right)}^{11}]$

$=[1+{}^{11}{c}_{1}x+{}^{11}{c}_2{x}^2+\cdots ][1+{}^{11}{c}_1{x}^2+{}^{11}{c}_2{x}^4+\cdots ]$

$={}^{11}{c}_0\cdot {}^{11}{c}_2\cdot x^4+{}^{11}{c}_2\cdot {}^{11}{c}_1\cdot x^4+{}^{11}{c}_4{}^{11}{c}_0{x}^4$

$=990x^4$

Option $\left(A\right)$