Is there a quick, fancy, way of proving sums such as this?
Prove that: $$\sum_{x=0}^{n} (-1)^x {n \choose x} = 0$$
A recent homework assignment I turned in had a couple problems similar to the above. For the most part, I used a proof by induction to solve them. They take forever to write up that way and I was wondering if there were any manipulations possible to speed these types of problems up.
Any suggestions would be greatly appreciated!