Prove the identity $\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-\binom{n}{3}+...=0$ using a combinatorial interpretation of the positive and negative terms.
While I know about Pascal's triangle/binomial coefficients from other classes I've taken, we have not learned it yet in this course, so I can't use row symmetry. The other thing I would think to use would be a bijection, but as far as practicalities go, I am unsure of how to set it up.
Also, we have already learned how to prove this using the binomial theorem, so I can't use that to solve this.
The sum would be $$\sum_{k=0}^n (-1)^k \binom{n}{k}$$
This is the number of ways to choose an even amount of objects from a set of $n$ minus the number of ways to choose an odd amount of objects. If you have already chosen/not chosen the previous objects, all that matters is whether the last object is chosen (for parity). Since there is one option for making it even, and one option for making it odd, the number of ways to choose evenly will be the same as the number of ways to choose oddly.