Calculating the sum of a binomial coefficient series

Question

Calculate this:

$$\bigl(\begin{smallmatrix} 80 \\0 \end {smallmatrix}\bigr)-\bigl(\begin{smallmatrix} 80 \\1 \end {smallmatrix}\bigr)+\bigl(\begin{smallmatrix} 80 \\2 \end {smallmatrix}\bigr)-\bigl(\begin{smallmatrix} 80 \\3 \end {smallmatrix}\bigr)+...-\bigl(\begin{smallmatrix} 80 \\79 \end {smallmatrix}\bigr)+\bigl(\begin{smallmatrix} 80 \\80 \end {smallmatrix}\bigr)$$

Can I get hints/suggestions for this? All I know is that: $\bigl(\begin{smallmatrix} 80 \\0 \end {smallmatrix}\bigr)$ and $\bigl(\begin{smallmatrix} 80 \\80 \end {smallmatrix}\bigr)$ must be equal to 1, right?

I know there must be some kind of trick but I just can't find it.

Thanks

Answer 1

Hint: Expand (for any positive integer $n$) $$(1 - 1)^n$$ with the Binomial Theorem.

This gives $$0 = (1 - 1)^n = \sum_{i = 0}^n (-1)^i {{n}\choose{i}} = {{n}\choose{0}} - {{n}\choose{1}} + \cdots + (-1)^{n - 1} {{n}\choose{n - 1}} + (-1)^{n} {{n}\choose{n}}.$$