Question

Let the coeffcient of $x^4$ in the expansion of${(2-x+3x^2)}^6$ be $K$. Find sum of digits of $K$.

Answer 1

The coefficient of $x^4$ can be achieved in following ways
$(-x{)}^4(3x^2{)}^0{2}^2$ or $(-x{)}^2(3x^2{)}^1{2}^3$ or $(-x{)}^0(3x^2{)}^2{2}^4$
Hence the required coefficient is
$2^2[\frac{6!}{4!.2!}+6\frac{6!}{1!.2!.3!}+36\frac{6!}{2!.4!}]$
$=4[15+360+540]$
$=4\left[915\right]$
$=3660$.