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 $-15360$.

By the multinomial formula,
$$(x^2-x+2)^{10}=\sum_{a+b+c=10}\begin{pmatrix}10!\\\!a!\,b!\;c!\!\end{pmatrix}x^{2a}(-1)^bx^b 2^c,$$
and obtaining $x^3$ means $2a+b=3$, whence $c-a=7$, so the possibilities aren't so many:


*

*either $a=0$, so $b=3$, $c=7$;

*or $a=1$, so $b=1$, $c=8$.


Therefore,  the  coefficient will be
$$ -2^7\biggl(\frac{10!}{3!\,7!}+2\frac{10!}{8!}\biggr)=-38\mkern 2mu400.$$ 
