The Simplex Lock Here's a problem that will weave its way in and out of many of the topics we discuss this summer. We'll start working on it right now, but don't feel that you have to get the "right answer" right away. We'll keep coming back to it, even after you might be satisfied that you know what's going on. There's a lot to discover in this problem. The Simplex company makes a combination lock that is used in many public buildings. It comes in several versions. Here are two: These 5-button devices are purely mechanical (no electronics). You can set the combination using the following rules: 1. A combination is a sequence of 0 or more pushes, each push involving at least one button. (There's one possible no-push combination: the door's just already unlocked. Not a great combination, but it counts.) 2. Each button may be used at most once (once you press it, it stays in). 3. Each push may include any of the buttons that haven't been pushed yet, up to and including all remaining buttons. 4. The combination does not need to include all buttons. 5. When two or more buttons are pushed at the same time, order doesn't matter.Artist's rendition of a Simplex lock. Here are some possible combinations: . {{1,3). (4}} . {{1, 2, 4). (3, 5}} * {{3}, (1,2}} . {{1, 2). (3} } { {1, 2, 3, 4,5} } . {(2). (1}, (3}} . ((1,2), (4). (3.5}} . (2) (Notation: {{1, 2), {3}} means "press 1 and 2 together, then press 3.") The company advertises thousands of combinations, and (as we say to our students) the question is, "Is the company telling the truth?" How many combinations are there on a 5-button Simplex lock! 2