Question

In the expansion of ${\left(1+x+x^2+x^3\right)}^{10}$ , the coefficient of $x^7$ is

• A. $9236$
• B. $9240$
• C. $9120$
• D. $10400$

Answer 1

Answer: (B)

${\left(1+x+x^2+x^3\right)}^{10}=(1+x{\left(1+x^2\right)}^{10}$
$\begin{array}{cccccc}(1+x{)}^{10}& {\left(1+x^2\right)}^{10}& \text{ coefficient }& \text{ from }(1+x{)}^{10}& \text{ coefficient }& \text{ from }{\left(1+x^2\right)}^{10}\\ 7& 0& {}^{10}{C}_7& {}^{10}{C}_0& & \\ 5& 2& {}^{10}{C}_5& {}^{10}{C}_1& & \\ 3& 4& {}^{10}{C}_3& {}^{10}{C}_2& & \\ 1& 6& {}^{10}{C}_1& {}^{10}{C}_3& & \end{array}$
Coefficient of $x^7=120\times 1+252\times 10+120\times 45+10\times 120=9240$