A student owns 6 different reference books and wants to carry some of them to the reading room, taking at least one. In how many different ways can she choose which books to carry?

Each of the 6 books is either carried or left behind — 2 choices per book, made independently of the others — so there are $2^6=64$ possible bundles. One of those is the empty bundle, which 'at least one' rules out, so the answer is 64 − 1 = 63. Trap 64 keeps that empty bundle. Trap 720 = 6! arranges all six books in a row, which answers a different question — here only WHICH books go matters, not the order they are stacked in. Trap $12=6\times 2$ adds the two possibilities book by book instead of multiplying them.