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Factorial and counting seat arrangements

Video transcript
- So, let's say that we have a round table and we have three chairs around that round table. This is one chair right over here, this is another chair, and that is another chair. We can number the chairs; that is chair one, that is chair two, and that is chair three. Now let's assume that there are three people who want to sit in these three chairs, so there is Person A, there is Person B, and there is Person C. What we want to do is we want to count the number of ways that these three people can sit in these three chairs. What do I mean by that? Well, if A sits in chair one, B sits in chair three, and C sits in chair two, that is one scenario right over there. So this is one of the possible scenarios, A in seat one, C in seat two, B in seat three. Another scenario could be B in seat one, maybe C still sits in seat two, C still sits there, and now A would be in seat three. So this is one scenario and two scenarios. My question to you is how many scenarios are there like this? I encourage you to pause the video and try to think through it. I'm assuming you've had a go at it, now let's work through it together. I want to do it very systematically so that we don't forget any of the scenarios. The way I'm going to do it is, I'm going to create three blanks that represent each of the chairs. So here, it's in a circle, but we could just say that blank is chair one, that's the blank for chair two, and that is the blank for chair three. I'm going to start with seat one, and I'm going to say, "Well, what are the different scenarios?" A could sit there, B could sit there, C could sit there. Then for each of those, figure out who could sit in seat two and then for each of those, figure out who could sit in seat three. So let's do that. First, what are the scenarios of who could sit in seat one? Maybe I'll write it this way, maybe I'll do seat one. So right now, we're only going to fill seat one. Well A could sit in seat one, in which case we haven't filled seat two or seat three yet. So that's two or three. B could sit in seat one, and we haven't figured out who sits in seat two or seat three yet. Then we could have C sitting in seat one, and we still have to figure out who's going to sit in seat two and seat three. For each of these, let's figure out who could be sitting in seat two. So, seats one and two. For this one right over here, we have A in seat one for sure. We could either have B sitting in seat two, in which case, I'll leave this blank although you can use a little bit of deductive reasoning to figure out who's going to be sitting in seat three. So I'll leave seat three blank. Or, you could have C sitting in seat two. We still haven't figured out who's sitting in seat three, although once again, a little bit of deductive reasoning might tell us. So these are the scenarios where A is in seat one. Now what about when B is in seat one? We could put A in seat two, and we still need to figure out who's in seat three. Or, we could put C in seat two, and we still need to figure out who's going to be in seat three. Finally, let's look at C, where C is in seat one. You could either put A in seat two or you could put B in seat two. Now let's fill out all three of the seats. So seats one, two, and three. For this scenario right over here, there's only one option that you could put in seat three, the only person who hasn't sat down yet is person C. So this will be A, B, C. And what would this be? The only person who hasn't sat down in this scenario yet to fill this chair, the only option for this chair, there's only one option here, it's going to be person B. It's A, C, B. This one is going to be B, A, C. C's the only person who could sit there, A is the only person who could sit there. B, C, A. And then here, B is the only person who could sit in seat three. And here A is the only person who could sit in seat three. How many scenarios do we have? We have one, two, three, four, five, and six. And six is our answer. That's the number of scenarios that you could have the different people sitting in the different chairs. Now, I know what you're thinking, "Okay Sal, this was a scenario with three people "and three chairs and I might've been able just do this "even if I didn't do it systematically. "But what if I have many many more chairs? "What if I had 60 chairs?" Well, the number of scenarios would get fairly large. What if I had five people and five chairs? Even this method right over here would take up a lot of paper space, or a lot of screen space. What do we do in that scenario? And the realization here is just thinking about what happened here. How many people could sit in seat one? If we're going seat by seat, how many possibilities were there to put in seat one? If you're seating in order, if you haven't filled any of the seats yet, right at the beginning, there's three possible people who could sit in seat one. You see it right over here, one two, three, because there's three possible people. For each of those, how many people could sit in seat two? For each of them, let's say the scenario where A is sitting in seat one, there's two possibilities for who could sit in seat two. Let me do this in a different color. So this, right over here. For each of these three, there's two possibilities of who could sit in seat two. If A is sitting in seat one, you could have either B or C in seat two. If B is sitting in seat one, you have either A or C for seat two. So for each of these three, you have two possibilities of who could sit in seat two. So you have six possibilities where you're filling in three times two, or you're filling in the scenarios for seats one and two. And then for each of those, how many scenarios are who can sit in seat three? For each of those, there was only one possibility for who could sit in seat three. We see it right there, that's because there's only one person left who hasn't sat down yet. So how many total possibilities are there? Well, three times two times one is equal to six. And so if we were to do that same exact thought exercise, with say, five seats, let's go through that exercise. It's interesting. So if we have one, two, three, four, five. So this is five seats with five people sitting down, and we want to figure out how many scenarios are there for all the different people and all the different seats. For the first seat, there's five different people who could sit in it. For each of those scenarios, there's four people who haven't sat down yet who could sit in the second seat. For each of those scenarios, there's going to be three people who haven't sat down yet, so those are the people who could sit in the third seat if we're filling out the seats in this order. For each of all of these scenarios, there's two people left who haven't sat down yet, who could sit in the fourth seat. And then for all of these scenarios, there's only one person left who hasn't sat down who would have to sit in the fifth seat. If you had this exact same thing, but instead of three chairs you had five chairs, the number of scenarios would be five times four times three times two times one. Which is what, 20 times six, which is equal to 120 scenarios. And so you see that the scenarios grow fairly quickly. And if you're wondering, "Hey, this is kind of a neat mathematical thing. Three times two times one." Or five times four times three times two times one. If you start with a number and you multiply that number times number one less than that all the way down to getting to one, that seems like neat kind of fun mathematical operation. And lucky for you, or maybe unlucky for you, because this might have been your chance at fame, this operation has already been defined. It's called the factorial operation. So this thing right over here, this is the same thing as three factorial. You write this little exclamation mark. This right over here, this is the same thing as five factorial. In general, if I said six factorial, that would be equal to six times five times four times three times two times one. Hopefully you enjoyed that.