Question

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

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

Answer 1

The correct option is D $310$

${(1+x+x^3+x^4)}^{10}$

$={\left[(1+x)(1+x^3)\right]}^{10}$

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

$\therefore$ Coefficients of $x^4=\left({}^{10}{C}_1\right)\left({}^{10}{C}_1\right){+}^{10}{C}_4$

$=100+\frac{\mathrm{10.9.8.7}}{\mathrm{1.2.3.4}}=100+210=310$