Answer
See proof
Work Step by Step
We have to prove the identity:
$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-...+(-1)^n\binom{n}{n}=0$
We have:
$(1-1)^n=0^n$
$(1-1)^n=0$
Use the Binomial Theorem to expand the left side:
$\binom{n}{0}1^n(-1)^0+\binom{n}{1}(-1)^{n-1}1^1+\binom{n}{2}(-1)^{n-2}1^2+.....+(-1)^n\binom{n}{n}1^01^n=0$
$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-...+(-1)^n\binom{n}{n}=0$
We proved the given identity.
