Subset-of-a-Subset and Binomial Identities

The subset-of-a-subset identity $\binom{n}{r}\binom{r}{k}=\binom{n}{k}\binom{n-k}{r-k}$ lets you factor a fixed $\binom{n}{k}$ out of any sum over the middle index, after which the leftover collapses to a clean row sum such as $2^{n-k}$.