The binomial coefficients add up to 2ⁿ

Show that for every integer $n\ge 0$,
$$\left(\genfrac{}{}{0ex}{}{n}{0}\right)+\left(\genfrac{}{}{0ex}{}{n}{1}\right)+\cdots +\left(\genfrac{}{}{0ex}{}{n}{n}\right)=2^n.$$

Try it three ways: from the binomial theorem, by counting the subsets of an $n$-element set, and by induction using Pascal's rule. One of them will feel like an explanation and the others like a check.