What is the coefficient of $x^3$ in expansion of $(x^2 - x + 2)^{10}$ 
What is the coefficient of $x^3$ in expansion of $(x^2 - x + 2)^{10}$.

I have tried:  
$$\frac{10!}{(3!\times7!)} \times (-x + 2)^7 \times (x^2)^3  $$
But got an incorrect answer $-15360$.

The terms in the development are of the form  $x^{2a}(-x)^b2^c\propto x^3$, and the exponents must satisfy
$$\begin{cases}a+b+c=10,\\2a+b=3.\end{cases}$$
The solutions are $0,3,7$ and $1,1,8$.
Hence by the multinomial formula
$$\frac{10!}{0!3!7!}1^0(-1)^32^7+\frac{10!}{1!1!8!}1^1(-1)^12^8=-120\cdot128-90\cdot256=-38400.$$

By Barry's method, condensed,
$$(p^{10})'=10p^9p',\\(p^{10})''=90p^8p'^2+10p^9p'',\\(p^{10})'''=720p^7p'^3+270p^8p'p''$$
(there is no $p'''$), gives, evaluating at $0$,
$$\frac{720\cdot2^7\cdot(-1)^3+270\cdot2^8\cdot(-1)\cdot2}6=-38400.$$
