[Math] How to begin proving this binomial coefficient identity: ${n\choose 0} – {n\choose 1} + {n\choose 2} – {n\choose 3} + \dots = 0$

Question

This is a homework question.

I'm asked to prove the identity:

$${n\choose 0} – {n\choose 1} + {n\choose 2} – {n\choose 3} + \dots = 0$$

(The sum ends with ${n\choose n} = 1$, with the sign of the last term depending on the parity of n.)

I recognize that the sequence:
$${n\choose 0}, {n\choose 1}, {n\choose 2}, {n\choose 3}$$

corresponds to the binomial coefficients. That is, if I choose $n = 5$, I get the sequence $1, 5, 10, 10, 5, 1$.

Working this out (or just looking at Pascal's triangle), it's obvious that this theorem is true. It looks like the triangle / the binomial coefficients are "symmetric", and so if you add one and subtract one and keep going, it's evident they will cancel out to be zero.

But how do I prove this? Is there a set way? Are there multiple methods for proving this? How should I get started, or what are some names of proving methods I should look into to begin?

Answer 1

Expand $(1-1)^n$ using the binomial theorem.