Question: At a fancy restaurant n students check their backpacks. They are the only ones to check backpacks. A child visits the checkroom and plays with the check tickets for the backpacks so they are all mixed up. If there are 5 students named Judy, Sam, Pat, Jill, and Jo, in how many ways may the backpacks be returned so that Judy gets her own backpack (and maybe some other students do, too)? What is the probability that this happens? What is the probability that Sam gets his backpack (and maybe some other students do, too)? What is the probability that Judy and Sam both get their own backpacks (and maybe some other students do, too)? For any particular two element set of students, what is the probability that these two students get their own backpacks (and maybe some other students do, too)? What is the probability that at least one student gets his or her own backpack? What is the probability that no students get their own backpacks? What do you expect the answer will be for the last two questions for n students? This classic problem is often stated using hats rather than backpacks (quaint, isn't it?), so it is called the hatcheck problem. It is also known as the derangement problem; a derangement of a set being a one-to-one function from a set onto itself (i.e., a bijection) that sends each element to something not equal to it.