Calculating a binomial sum

Question

I came across this excercise in an old exam (in discrete math), and I don't know how to approach it: $$\sum_{k=0}^{10}\left(\frac{1}{2}\right)^k\left(-1\right)^k\binom{10}{k}$$ I know the answer is $2^{-10}$, but I don't know why. When I was going through my text book I saw something similar regarding Catalan numbers generating functions. Thanks!

Answer 1

Hints: $(-1)^k=(-1)^{10-k}$ and binomial theorem.

Answer 2

Another Hint

$$\sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k} = (x+y)^n$$