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in this video we are going to introduce ourselves to the idea of permutations which is a fancy word for a pretty straightforward concept which is what are the number of ways that we can arrange things how many different possibilities are there and to make that a little bit tangible let's have an example with say a sofa my sofa can seat exactly 3 people I have seat number one on the left of the sofa seat number two in the middle of the sofa and seat number three on the right of the sofa and let's say we're going to have three people who are going to sit in these three seats person a person B and person C how many different ways can these three people sit in these three seats pause this video and see if you can figure it out on your own well there are several ways to approach this one way is that just trying to think through all of the possibilities you could do it systematically you could say all right let's if I have person a in seat number one then I could have person B in seat number two and person C in seat number three and I could think of another situation if I have person a and seat number one I could then swap B and C so it could look like that and that's all of the situations all of the permutations where I have a in seat number one so now let's put someone else in seat number one so now let's put B in seat number one and I could put a in the middle and C on the right or I could put B in seat number one and then swap a and C so C and then a and then if I put C in seat number one well I could put a in the middle and B on the right or with C in seat number one I could put B in the middle and a on the right and these are actually all of the permutations and you can see that there are 1 2 3 4 5 6 now this wasn't too bad and in general if you're thinking about permutations of 6 things and or three things and three spaces you can do it by hand but it could get very complicated if I said hey I have a hundred seats and I have a hundred people that are going to sit in them how do I figure it out mathematically well the way that you would do it and this is going to be a technique that you can use for really any number of people and any number of seats is to really just build off of what we just did here well we did here is we started with seat number one and we said all right how many different possibilities are who how many different people could sit in seat number one assuming no one has sat down before well three different people could sit in seat number one you can see it right over here this is where a is sitting in seat number one this is where B is sitting in seat number one and this is where C is sitting in seat number one now for each of those three possibilities how many people can sit in seat number two well we saw when a sits in seat number one there's two different possibilities for seat number two when B sits in seat number one there's two different possibilities for seat number two when C sits in seat number one this is a tongue-twister there's two different possibilities for seat number two and so you're gonna have two different possibilities here another way to think about it is one person has already sat down here there's three different ways of getting that and so there's two people left who could sit in the second seat and we saw that right over here where we really writ wrote out the permutations and so how many different permutations are there for seat number one and seat number two well you would multiply for each of these three you have two for each of these three and seat number one you have two in seat number two and then what about seat number three well if you know who's in seat number one and seat number two there's only one person who can be in seat number three and another way to think about it if two people have already sat down there's only one person who could be in seat number three and so mathematically what we could do is just say three times two times one and you might recognize the mathematical operation factorial which literally just means hey start with that number and then keep multiplying it by the numbers one less than that and then one less than that all the way until you get to 1 and this is 3 factorial which is going to be equal to 6 which is exactly we've got here and to appreciate the power of this let's extend our example let's say that we have five seats 1 2 3 4 5 and we have 5 people person a B C D and E how many different ways can these five people sit in these five seats pause this video and figure it out well you might immediately say well that's going to be five factorial which is going to be equal to 5 times 4 times 3 times 2 times 1 5 times 4 is 20 20 times 3 is 60 and then 60 times 2 is 120 and then 120 times 1 is equal to 120 and once again that makes a lot of sense there was 5 different if no one's sat down there's 5 different possibilities for seat number 1 and then what for each of those possibilities there's 4 people who could sit in seat number 2 and then for each of those 20 possibilities in seat numbers 1 & 2 well there's going to be three people who could sit in seat number 3 and for each of these 60 possibilities there's two people who can sit in seat number 4 and then once you know who's in the first four seats you know who has to sit in that fifth seat and that's where we got that 120 from