Calulating coefficiencies

Question

I'm trying to calculate the number ofcoefficiencies of every $x^k$
in the expansion of $(x-\frac{1}{x})^{100}$ for an arbitrary $k\in\mathbb{Z}$

Now I tried the following formula:
$\binom{100}{k}$ eventhough similar to the demonstration,
Something is missing here (and/or the lack of explanation makes it harder to understand).
help?

Answer 1

Hint: $$\left(x-\frac{1}{x}\right)^{100}=x^{100}\left(1+(-x^{-2})\right)^{100}.$$ Now try the binomial theorem on the right-hand side.