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CCSS.Math:

the first time you're exposed to permutations and combinations it takes a little bit to get your brain around it so and I think it never hurts to do as many examples but each incremental example I'm gonna go I'm gonna review what we've done before but hopefully go a little bit further so let's just take another example and this isn't the same thing and in the next and videos after this I'll start using other examples other than just people sitting in chairs but let's stick with it for now so let's say we have six people again so person a b c d e and f so we have six people and now let's put them into four chairs and we can go through this fairly quickly one two three four chairs and we've seen this show multiple times how many ways how many permutations are there of putting these six people into four chairs well the first chair if we see them in order we might as well we could say well there'd be six possibilities here now for each of those six possibilities there would be five possibilities of who we put here because one person's already sitting down now for each of these thirty possibilities of seating these first two people there'd be four possibilities of who we put in chair number three and then for each of these what is this 120 possibilities there would be three possibilities of who we put in chair four and so this 6 times 4 times 6 times 5 times 4 times 3 is the number of permutations and we've seen in one of the early videos on permutations that or when we talk about the permutation formula one way to write this if we wanted to write it in terms of factorial we could write this as as six factorial six factorial which is going to be equal to 6 times 5 times 4 times 3 times 2 times 1 but we want to get rid of the 2 times 1 so we're going to divide that we're gonna divide that now what's 2 times 1 well 2 times 1 is 2 factorial and where did we get that well we wanted the first for the first 4 factors of 6 factorial so if you want and that's where the 4 came from we wanted the first 4 factors and so the way we've got 2 is we said 6-4 6-4 that's going to get us what we want to get that that's going to give us the number that we want to get rid of all right so we wanted to get rid of two or the factors we want to get rid of so that's going to give us two factorial so if we use six minus four factorial then that's going to give us two factorial which is two times one and then these cancel out and we are all set and so this is one way this is you know I put in the particular numbers here but this is a review of the permutations formula where people say hey if I'm saying and if I'm thinking N Things and I want to figure out how many permutations are there putting them into let's say K spots it's going to be equal to n factorial over n minus K factorial that's exactly what we did over here where six is N and K or four is K four is K actually let me color code the whole thing so that we see so that we see the so that we see the parallel now all of that is review but then we went into the world of combinations and in the world of combinations we said okay permutations permutations make a difference between who's sitting in what chair so for example in the permutations world this is all review we've covered this in the first combinations video and the permutations world a b c d and d a b c these would be two different permutations that's being counted in whatever number of the Cistus is what this is 30 times 12 this is this is equal to this is equal to 360 so this is each of these this is one permutation this is another permutation and if we keep doing it we would count up to 360 but we learned in combinations when we're thinking about combinations let me write combinations so if we're saying and choose and choose K or how many combinations are there if we take K things and we just want to figure out how many combination so if we start with n if your pool of N Things and we want to say how many combinations of K things are there then we would count these as the same combination so what we really want to do is we want to take the number of permutations there are we want to take the number of permutations there are which is equal to and factorial over n minus K factorial over n minus K factorial and we want to divide by the number of ways that you could arrange for people once again and this takes I remember the first time I learned it took my brain a little while so if it's taking a little while to think about it not a big deal it's it's it can be confusing at first but it'll hopefully if you keep thinking about it hopefully you will see clarity at some moment but what we want to do is we want to divide by all of the ways that you could arrange for things because once again and the permutations it's counting all of the different arrangements of four things but we don't want to count all of those different arrangements of four things we want to just say well they're all one combination so we want to divide by the way the number of ways to arrange four things now if or the number of ways to arrange K things so let me write this down so what is the number of ways number of ways to arrange K things K things in K spots and I encourage you to pause the video because this is actually a review from the first permutation video well if you have K spots let me do it so this is the first spot the second spot third spot and then you're gonna go all the way to the cait spot well for the first spot there could be K possibilities there's K things that could take the first spot now for each of those K possibilities how many things could be in the second spot well it's gonna be K minus one cuz you already put you already put something in the first spot and then over here was it gonna be K minus two all the way to the last spot there's only one thing that could be put in the last spot so what is this thing here K times K minus 1 times K minus 2 times K minus 3 all the way down to 1 well this is just equal to K factorial the number of ways to arrange K things in K spots K factorial the number of ways to arrange four things in four spots that's four factorial the number of ways to arrange three things in three spots it's three factorial so we could just divide this we could just divide this by K factorial and so this would get us this would get us n factorial divided by K factorial K factorial x times n minus K factorial and minus K and minus K and I'll put the factorial right over there and this right over here is the formula this right over here is the formula for combinations sometimes this is also called the binomial coefficient people will call this n choose K they'll also write it like this and choose K especially when they're thinking in terms of binomial coefficients but let's I got into kind of an abstract tangent here when I started getting into the the general formula but let's go back to our example so in our example in our example we saw there was a 360 ways of seating six people into four chairs but what if we didn't care about who's sitting in which chairs and we just want to say how many ways are there to choose four people from a pool of six well that would be that would be how many ways are there so that would be six how many combinations if I'm starting with a pool of six how many combinations are there how many combinations are there four selecting four or another way of thinking about it is how many ways are there to from a pool of six items people in this example how many ways are there to choose four of them and that is going to be you know we could do it and then I'll reason through it and like I always say I don't I'm not a huge fan of the formula every time I you know I revisited after a few years I actually just reread think about it as opposed to memorizing it because memorizing is a good way to not understand what what's actually going on but if we just applied the formula here but I really understand I really want you to understand what's happening with the formula it would be six factorial over four factorial over four factorial times 6 minus 4 factorial six whoops let me actually let me just so this is 6 - 4 factorial so this part right over here 6 - 4 or 5 actually let me write it out because I know this can be a little bit confusing the first time you see it so 6 minus 4 factorial factorial which is equal to which is equal to 6 factorial over 4 factorial over 4 factorial times this thing right over here is 2 factorial times 2 factorial which is going to be equal to we could just write out the factorial 6 times 5 times 4 times 3 times 2 times 1 over 4 4 times 3 times 2 times 1 times times 2 times 1 and of course that's going to cancel with that and then the 1 really doesn't change the value so let me get rid of this one here and then let's see this 3 can cancel with this 3 this 4 could cancel with this 4 and then it's 6 divided by 2 is going to be 3 and so we are just left with 3 times 5 so we are left with we are left with there's 15 combinations there's 360 permutations for putting 6 people into 4 chairs but there's only 15 combinations because we're no longer counting all of the different arrangements for the same four people in the four chairs we're saying hey if it's the same four people that is now one combination and you could see how many ways are there to arrange four people into four chairs well that's the 4 factorial part right over here the 4 factorial part right over here which is 4 times 3 times 2 times 1 which is 24 so we just essentially just took the 360 divided by 24 to get 15 but once again I don't want to strike him I don't think I can stress this enough I want to make it clear where this is coming from this right over here let me circle this piece right over here is the number of permutations and this is really just so you can get to 6 times 5 times 4 times 3 it was exactly what we did up here reason through it and then we just want to divide by the number of ways you can arrange for items in four spaces