Question

The coefficient of $x^5$ in the expansion of ${(2-x+3x^2)}^6$ is

• A. $-5049$
• B. $5051$
• C. $-5052$
• D. $5050$

Answer 1

The correct option is C $-5052$

$\frac{6!}{p!q!r!}{\left(2\right)}^p{(-x)}^q{\left(3x^2\right)}^r$
where $p+q+r=6$ and $q+2r=5$

Cases :
$\left(1\right)r=0,p=1,q=5$
$\left(2\right)r=1,p=2,q=3$
$\left(3\right)r=2,p=3,q=1$

$\therefore$ Coefficient $=\frac{6!}{1!5!0!}\times 2^1\times {(-1)}^5\times 3^0+\frac{6!}{2!3!1!}\times 2^2\times {(-1)}^3\times 3^1+\frac{6!}{3!1!2!}\times 2^3\times {(-1)}^1\times 3^2$

$=-12-720-4320=-5052$