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. $Noneofthese$

Answer 1

The correct option is A $990$

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

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

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

$=({}^{11}{C}_0+{}^{11}{C}_{1}x+{}^{11}{C}_2{x}^2+.....+{}^{11}{C}_{11}{x}^{11})({}^{11}{C}_0+{}^{11}{C}_1{x}^2+{}^{11}{C}_2{x}^4+.....+{}^{11}{C}_{11}{x}^{22})$

Therefore,

Coefficient of $x^4={}^{11}{C}_4\cdot {}^{11}{C}_0+{}^{11}{C}_0\cdot {}^{11}{C}_2+{}^{11}{C}_2\cdot {}^{11}{C}_1=330+55+605=990$