You can do it in a sloppy way .....
$\displaystyle {-1 \choose k} = \frac{(-1)!}{k!(-1 - r)!}$
Apply the factorial dogmatically:
$\displaystyle = \frac{(-1)(-2)(-3) .... (-k)(-k-1)!}{(-1 - k)! k!}$
Cancel the bad stuff:
$\displaystyle = \frac{(-1)^k k!}{k!} = ....$
And you're left with the good stuff.
I imagine a formal approach would use the gamma function (with great care since $\displaystyle x = -n$, $\displaystyle n \in \mathbb{Z}$, are simple poles .....) Maybe if someone has the time and the inclination (but I don't think a pre-algebra and algebra forum is the place) .....