Question

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

• A. $990$
• B. $780$
• C. $1040$
• D. $520$

Answer 1

The correct option is A $990$

Coefficient of $x^4$ in ${(1+x+x^2+x^3)}^{11}$ is

$=$ coefficient of $x^4$ in ${(1+x)}^{11}.{(1+x^2)}^{11}$

$=$ (coefficient of $1$ in $(1+x^{11}))$.(coefficient of $x^4$in ${(1+x^2)}^{11}$)+(coefficient of $x^{2}in{(1+x)}^{11})$.(coefficient of $x^2$ in ${(1+x^2)}^{11}$)+(coefficient of $x^4$ in ${(1+x)}^{11})$.(coefficient of $1$ in ${(1+x^2)}^{11}$)

$=\left(1.\left(\frac{11}{12}\right)\right)+(\left(\frac{11}{2}\right).\left(\frac{11}{1}\right))+\left(\left(\frac{11}{12}\right).1\right)$

$\frac{11.10}{2!}+11.\frac{11.10}{2!}+\frac{\mathrm{11.10.9.8}}{4!}=55+605+330$

$=990$