Prove that $\sum _{k=0}^{n}{{(-1)}^{k}{\binom n k}}=0$

Question

Prove that: $$\sum _{k=0}^{n}{{(-1)}^{k}\binom n k}=0$$

I tried with induction and failed.

A solution explain would be greatly appreciated.

Answer 1

The Binomial Theorem says $$(a+b)^n=\sum_{0\le r\le n}\binom nr a^{n-r}b^r$$

for real or complex $a, b$, and non-negative integer $n$.

Put $a=1,b=-1$