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 $-15360$.

You can get $x^3$ only by multiplying $x^2$ with $-x$, or by multiplying three $(-x)$'s together where the rest of the terms you are multiplying with should be $2$'s.
We can multiply $x^2$ with $(-x)$ in $2\cdot \binom{10}{2}$ ways (We multiply by $2$ to take into account the order of $x^2$ and $(-x)$). And we can multiply three $(-x)$'s together in $\binom{10}{3}$ ways. Since both these yield a $(-1)$ coefficient, and taking into account that the rest of the terms are $2$'s, we get
$$-2^8\cdot 2\binom{10}{2}-2^7\cdot\binom{10}{3} = -38400$$
