Shuffle a Deck and Check

Take a standard 52-card deck and shuffle it thoroughly. The arrangement in your hands is a permutation — one out of 52! possibilities. Yet you can prove the idea with far fewer items.

Grab 4 books and arrange them in every possible order. There are 4! = 24 arrangements, and you can run through all of them in a few minutes. Add just a few more books and the orderings rocket past what anyone could ever list by hand.

That is the surprise of factorial growth: the gap between "can enumerate" and "cannot" is just a handful of items.