Question

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

• A. $330$
• B. $990$
• C. $900$
• D. $895$

Answer 1

Answer: (B)

We have, $(1+x+x^2+x^3{)}^{11}=(1+x{)}^{11}(1+x^2{)}^{11}$
$=(1+{}^{11}{C}_{1}x+{}^{11}{C}_2{x}^2+{}^{11}{C}_3{x}^3+{}^{11}{C}_4{x}^4+...)\times$
$(1+{}^{11}{C}_1{x}_2+{}^{11}{C}_2(x^2{)}^2+...)$
$\therefore$ Coeffficient of $x^4={}^{11}{C}_2+{}^{11}{C}_2\times {}^{11}{C}_1+{}^{11}{C}_4$
$=55+605+330=990$