Question

In the expansion of $(1+x+x^3+x^4{)}^{10},$ the coefficient of $x^4$ is

• A. ${}^{40}{C}_4$
• B. ${}^{10}{C}_5$
• C. $210$
• D. $310$

Answer 1

The correct option is D $310$

$(1+x+x^3+x^4{)}^{10}=((1+x)+x^3(1+x){)}^{10}$
$=(1+x{)}^{10}(1+x^3{)}^{10}$
By applying binomial expansion, we get
$=(1{+}^{10}{C}_{1}x{+}^{10}{C}_2{x}^2{+}^{10}{C}_3{x}^3{+}^{10}{C}_4{x}^4...)(1{+}^{10}{C}_1{x}^3{+}^{10}{C}_2{x}^6+...)$
Therefore, coefficient of $x^4$is${}^{10}{C}_1{}^{10}{C}_1{+}^{10}{C}_4=100+210=310$