Use the Binomial Theorem to show that $0 = \sum_{k=0}^ n (-1)^{k} {n \choose k }$

Question

Use the Binomial Theorem to show that "$$0 = \sum_{k=0}^ n (-1)^{k} { n \choose k}$$".

Answer 1

Try expanding $(1-1)^n$, using the binomial theorem.

Answer 2

$$(1-1)^n=\sum_{x=0}^n {n\choose x}(-1)^x=0$$

Conisder the following expansion $$(x-a)^n=\sum_{i=0}^n{n \choose i}x^{n-i}(-a)^i$$. . In your question $x =1$ and $a=1$.