Well, clearly, if we're picking them at random, for the first choice, for the facilitator, I have 25 choices. If we swap around the people into the different roles, then we have a different steering committee. So I'll just point out here that the order does matter. How many different possible steering committee could be chosen? So it sound like if Harry is chosen for the facilitator and Sally is chosen for the union rep, that would be different is Sally is chosen as the facilitator and Harry is chosen as the union rep. So it sounds like three different jobs for three different people. Okay, a small division of a company with 25 employees will choose a three person steering committee consisting of a facilitator, a union rep and a secretary. Pause the video, and then we'll talk about this. So just keep this in the back of your mind right now, we'll talk about this more formally and we'll have a special notation for it in a couple lessons. We will formalize this in a couple lessons when we discuss factorials. And in general, if we have to arrange N different items in order, the total number of orders the product of N times every positive integer less than N. So that would be 720 orders for any 6 distinct different items and no restrictions. Notice that in arranging any six different items in order, the total number of orders the product of six and every positive integer less than it. And that's the number of different orders in which we can put these books. Now we can simplify this a little bit, the 3 times 2 is 6, the 5 times 4 is 20. And so N will be 5 times 4 times 3 times 2 times 1. So I really want no choice at that point. And by the time I get to the last book, I'm only gonna have one choice because five books are already going to be in place, and I'm just gonna have that one last book. So, the third book, I'll have four choices, the second book I'll have three choices, The fifth book, I'll have two choices. And so forth on each stage there are fewer choices that I'll have because books have already been put in the slot. And so that first book is already sitting in the first slot, so when I go to make that choice of the second slot, I have 5 choices left, there still 5 books available that I could put in that second slot. Now here's the tricky part, for the second slot, I already picked a book. So when I start out, I have six books, I can pick up any one and put it in that first slot. Stage number one is what are we going to put in the first slot? Stage number two would be what are we going to put in the second slot, that sort of thing? Well, in the first slot, I have six choices. Suppose we have six different books that will place on a shelf, in how many different orders can we place these books? Well, think about it this way, the various places are stages. Well, 8 times 12 is 96, so 80 times 12 would have to be 960, that's the answer. Or 4 times 5, of course, is 20, then multiply by that other 4 that's 80. So the number of ways to do this would be 4 times 5 times 12 times 4. So we can just simply multiply the numbers, that's what the fundamental counting principle tells us. And there's no restrictions here, in other words, any salad can go with any appetizer, they can go with any entree, so there's no restriction at all. So of the meals like that, how many different possible meals are there? Well, the fundamental counting principles is perfect here because we're in stages, we can treat each course separately. And so the whole idea is that at any particular dinner chosen, you'll get a salad and appetizer and entree and dessert. So, for formal dinner, guests have the choice of 1 of 4 salad, so, choice of 1 of 4, 1 of 5 appetizers, 1 of 12 entrees, and 1 of 4 desserts. So I realized this is very abstract, now I'm gonna show a few examples. This is the fundamental counting principle. So the number of ways we could do it in the first stage times the number of choices we can make in the second stage times the number of choices we make in the third stage, etc. The total number of ways to do the task was simply be the product of all these numbers. Suppose the first stage can be done in n sub 1 ways, the second way and then sub 2 ways and so forth. Suppose we can divide a given task in two stages. Fundamental counting principle, Is a general way to approach tasks that can be broken into stages. And it really is based on the very simple idea that the word and means multiply. This is, as the name suggests, the single most important principle in this entire module.
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