PermutationsandCombinations_VFIN

PermutationsandCombinations_VFIN -...

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PermutationsandCombinations_VFIN.doc Permutations and Combinations or the Mathematics of Hugs and Kisses 1. An apology This is a topic that I use to discuss much earlier in the course, because I had the impression that it was very important in leading up to the discussion of probabilistic models and language models which we have already talked about. Eventually, I learned that those could be discussed without having talked about permutations and combinations at all, and this very interesting topic almost fell completely out of the course. However, I realized that it has some interesting applications in examining the complexity of networks of relations among people, and among documents. So it managed to fight its way back into the course, but a little bit later in the syllabus. 2. Permutations Let’s begin by thinking about permutations. The number of permutations that can be made with some number n of items means the number of different orders that we can place them in. 2.1. A simple example For example if we have 3 items that are named A, B and C We notice that while we’re still keeping A in the first position A B C We could switch the second two and get a new order like this. A B C But of course we could also have done that with B in the first position B A C And then: B C A And so we would get two more orders. And finally we could have done it with C in the first position getting two additional orders. C A B
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PermutationsandCombinations_VFIN.doc C BA Altogether we see that there are 6 different orders that can be made with those three different things. And it does not seem that there are any more. So we can write that the number of permutations of 3 objects is given by the equation 6 n P . 2.2. What happens if we add another object to the set? Now we are going to want to apply some kind of mathematical induction here so we’d like to ask the question: “if we already know a formula for n P can we calculate the number 1 n P ?” I will argue that we can. If we have one more item that we need to put into the list, let’s call it X, we see that for any one of the lists of orders that we have made, for example A C B, there are four different ways that we could put the X into the list. We could put it in front of any one of the letters; that makes 3 ways. Or we could put it after all of them and that’s the fourth way. That means that if we add a fourth item to the set of items, each of our old orders of the 3 items can be turned into 4 different ones. And it’s clear that all 4 of those will be different from each other. This means that we can write the equation 43 4 PP . Now, in fact, there’s nothing at all special about the fact that we went from 3 to 4; no matter how many items we had in the list if we were going to add one more there would be 1 n places that we could put it “into” the list. So we can actually write the equation that says that 1 ( 1) nn P n P  .
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