How many $10$-digit numbers use only the digits $0, 1, 2$ with each digit appearing at least twice or not at all?
I know I need the coefficient of $\frac{x^{10}}{10!}$ in:
$$\left(1+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\cdots \right)^3=(e^x-x)^3$$
but what do I do here?
You correctly expanded the RHS as $$(e^x)^3-3x(e^x)^2+3e^xx^2-x^3$$
Now write this as $$(e^{3x})-3x(e^{2x})+3e^xx^2-x^3$$
We look to the Taylor series of $e^{3x}$, found by substituting $y=3x$ in the Taylor series of $e^x$:
$$e^{3x} = 1+3x+\frac{3^2x^2}{2!} + \frac{3^3x^3}{3!} + \cdots + \frac{3^{10}x^{10}}{10!} + \cdots $$
Can you find those form $e^{2x}$ and $e^{x}$ yourself? If not, just leave a comment and I'll help you.