Question

Coefficient of $x^7$ in the expansion of ${(1-2x+x^3)}^6$ is

• A. $280$
• B. $360$
• C. $-450$
• D. $-280$

Answer 1

The correct option is B $360$

$x^7$ implies $\left(x^3{)}^2\right(-2x{)}^1(1{)}^3$
The coefficient being $=\frac{6!.(-2)}{2!3!1!}$
And $\left(x^3\right)(-2x{)}^4(1{)}^1$
The coefficient being $=\frac{6!.(-2{)}^4}{1!4!1!}$
Hence the coefficient of $x^7$ in the above expansion is
$\frac{6!.(-2)}{2!3!1!}+\frac{6!.(-2{)}^4}{1!4!1!}$
$=-120+480$
$=360$