Question

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

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

Answer 1

The correct option is D $310$

Given expression can be written as :

$(1+x{)}^{10}(1+x^3{)}^{10}$ $=(1{+}^{10}{C}_{1}x{+}^{10}{C}_2{x}^2{+}^{10}{C}_3{x}^3{+}^{10}{C}_4{x}^4+\dots )(1{+}^{10}{C}_1{x}^3{+}^{10}{C}_3{x}^6+\dots )$

The coefficient of $x^4$ is:

${}^{10}{C}_1{\times }^{10}{C}_1{+}^{10}{C}_4$

$=100+210=310$