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so let's keep thinking about different ways to sit multiple people in the certain number of chairs so let's say we have six people we have person a we have person B with person C person D person e and we have person F so we have six people and for the sake of this video we're gonna say we want to figure out all the scenarios all the possibilities all the permutations all the ways that we could put them into three chairs so that's chair number one chair number two and chair number three this is all review this is covered in the permutations video but it'll be very instructive as we move into a new concept so what are all of the permutations of putting six different people into three chairs well like we've seen before we could start with the first chair and we could say look if no it says if we haven't we haven't seated anyone yet how many different people could we put in chair number one well there's six different people who could be in chair number one in a different color there are six people who could be in chair number one six different scenarios for who sits in chair number one now for each of those six scenarios how many people how many different people could sit in chair number two well in each of those six scenarios we've taken one of the six people to sit in chair number one so that means you have five out of the six people left to sit in chair number two or another way to think about it is there's six scenarios of someone in ter number one and for each of those six you have five scenarios for who's in chair number two so you have a total of 30 scenarios where you have seated where you have seated six people in the first two chairs and now if you want to say well what about for the three chairs well for each of these thirty scenarios how many different people could you put in in chair number three well you're still gonna have four people standing up not in chairs so for each of these 30 scenarios you have four people who you could put in chair number three so your total number of scenarios or your total number of permutations where we care who's sitting in which chair is six times five times four which is equal to [Music] 120 permutations permutations now these permutations now it's worth thinking about what permutations are counting now remember we care when we're talking about permutations we care about who's sitting in which chair so for example for example this is one permutation and this would be counted as another permutation and this would be counted as one as another permutation this would be kind of that's another permutation so notice these are all with the same three people but we're putting them in different chairs and this counted that that's kind of in this hundred twenty I could keep going we could have that or we could have that so our thinking in the permutation world we would count all of these or we would count this as six different permutations these are going towards this 120 and of course we have other permutations where we involve other people where we have it could be f bc f CB f AC F F actually let me do it this way that would be a little bit more systematic F let me do it b b FC B C F and obviously I could keep going I do wonder twenty of these I'll do I'll do two more you could have C F B and then you could have C B F so in the permutation world these are these are literally twelve of the hundred twenty permutations but what if we all what if all we cared about is the three people we're choosing to sit down but we don't care in what order that they're sitting or which chair they're sitting so in that world these would all be one this is all this is all the same set of three people if we don't care which chair they're sitting in this would also be the same set of three people and so this question if I have six people sitting in three chairs how many ways can I choose three people out of the six where I don't care which chair they sit on and I encourage you to pause the video and try to think of what that number would actually be well a big clue was when we we essentially wrote all of the permutations where we've picked a group of three people we see that there's six ways of arranging the three people when you pick a certain group of three people that turned into six permutations and so if you all you want to do is care about well how many different ways are there to choose three from the from the six you would take your whole permutations you would take your number of permutations you would take your number of permutations and then you would divide it by the number of ways to arrange three people number of ways to arrange arrange three people and we see that you can arrange three people or even three letters you can arrange it in six different ways so this would be equal to 120 divided by six or this would be equal to twenty so there were 120 permutations here if you said how many different arrangements are there of taking six people and putting them into three chairs that's 120 but now we're asking another thing we're saying if we start with a hundred and twenty people and we want to choose and we want to say if we're starting if we're starting with six people and we want to figure out how many ways how many combinations how many ways are there for us to choose three of them then we end up with 20 combinations combinations of people this right over here once again this right over here is just one combination is the combination a B C I don't care what order they sit and I've chosen them I've chosen these three of the six this is a combination of people I don't care about the order this right over here is another combination it is F C and B once again I don't care about the order I just care that I've chosen these three people so how many ways are there to choose three people out of six it is 20 it's the total number of permutations it's 120 divided by the number of ways you can arrange three people