Question

Find the coefficient of $x^7$ in ${\left(1-x+2x^3\right)}^{10}$.

Answer 1

Answer: 960

Coefficient of $x^7$ in ${\left(1-x+2x^3\right)}^{10}$
general term : ${}^{10}{C}_r(1-x{)}^{10-r}\cdot {\left(2x^3\right)}^r;{}^{10}{C}_r(1-x{)}^{10-r}\cdot 2^r\cdot x^{3r}$.
When $r=0$, coefficient of $x^7$ in ${}^{10}{C}_0(1-x{)}^{10}$
$\Rightarrow -{}^{10}{C}_7$.
When $r=1$, coefficient of $x^4$ in ${}^{10}{C}_1(1-x{)}^9\cdot 2$
$\Rightarrow 20\left({}^9{C}_4\right)$
When $r=2$, coefficient of $x^1$ in ${}^{10}{C}_2(1-x{)}^8\cdot 2^2$
$\Rightarrow 180\left(-{}^8{C}_1\right)$
$\Rightarrow$ coefficent of $x^7=\left(-{}^{10}{C}_7+20\cdot {}^9{C}_4-180\cdot {}^8{C}_1\right)$
$=(-120,2520-1440)=960$.