Question

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

• A. ${}^n{C}_4$
• B. ${}^n{C}_4+{}^n{C}_2$
• C. ${}^n{C}_4+{}^n{C}_4+{}^n{C}_2$
• D. ${}^n{C}_4+{}^n{C_2}^n{C}_1+{}^n{C}_2$

Answer 1

Answer: (D)

${(1+x+x^2+x^3)}^n$
$={[(1+x)+x^2(1+x)]}^n$
$={(1+x)}^n{(1+x^2)}^n$
$=(1+{}^n{C}_{1}x{+}^n{C}_2{x}^2{+}^n{C}_3{x}^3{+}^n{C}_4{x}^4+...)$
$\times (1{+}^n{C}_1{x}^2{+}^n{C}_2{x}^4+...)$
$\therefore$ Coefficient of $x^4$
$={}^n{C}_2{+}^n{C_1}^n{C}_2{+}^n{C}_4$