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CCSS.Math:

we know that if we had five people let's say person a person B person C person D and person E and we wanted to put them in five different let's say positions or chairs so position one position to position three position four and position five if we wanted to count the number of scenarios or we could say the number of permutations of putting these five people in these five chairs well we could say well we have five different if we if we seated people in order which we might as well do we could say look five different people could sit in chair one so for each of those scenarios four different people could sit in chair two now for each of these scenarios now so we have 20 scenarios five five times four we have twenty scenarios where we've seated seat ones and seat 1 and C 2 how many people could we now seat in seat three for each of those 20 scenarios well 3 people haven't sat down yet so there's three possibilities there so now there's five times four times three scenarios for seating the first three people how many people are left for seat four well two people haven't sat down yet so there's two possibilities so now there's five times four times three times two scenarios of seating the first four seats and for each of those how many possibilities are there for the fifth seat well 1 / 4 each of those scenarios we only have one person who hasn't sat down left so there's one possibility and so the number of permutations the number of let me write this down the number of permutations permutations of seating these five people in five chairs is five factorial five factorial which is equal to five times four times three times two times one which of course is equal to C twenty times six which is equal to 120 and we've already covered this in a previous video but now let's do something maybe more interesting or maybe you might find it less interesting let's say that we still have let's say we have these five people but we don't have as many chairs so not everyone is going to be able to sit down so let's say that we only have three chairs so we have chair one we have chair two and we have chair three so how many ways can you have five people where only three of them are going to sit down in these three chairs and we care which chair they sit in and I encourage you to pause the video and think about it so I am assuming you have had your go at it so let's use the same logic so how many if we're if we seat them in order we might as well how many different people if we haven't seen sat anyone yet how many different people could sit in seat one well we could have no if no one sat down we have five different people well five different people could sit in seat one well for each of these scenarios where one person has already sat in seat one how many people could sit in seat two well in each of these scenarios if one person is sat down there's four people left who haven't been seated so four people could sit in seat two so we have five times four scenarios where we have seated seats one and C two now for each of those 20 scenarios how many people could sit in seat three well we haven't sat we haven't we haven't Seaton or set three of the people yet so for each of these twenty we could put three different people in seat three so that gives us five times four times three scenarios so this is equal to five times four times three scenarios which is equal to this is equal to 60 so there's 60 permutations of sitting five people in three chairs now this and this my brain you know whenever I start to think in terms of permutations I actually think in these ways I just literally draw it out because especially you know I don't like formulas I like to actually conceptualize and visualize what I'm doing but you might say hey you know when when we just did five different people and five different chairs and we cared which seat they sit in we had this five factorial you know factorial is kind of a neat little operation there how can I relate factorial to what we did to what we did just now well it looks like we kind of did factorial but then we stopped we stopped at we didn't go times two times one so one way that we one way to think about what we just did is we just did 5 times 4 times 3 times 2 times 1 but of course we actually didn't do the 2 times 1 so you could take that and you could divide by 2 times 1 and if you did that then this 2 times 1 would cancel with that 2 times so on and you'd be left with 5 times 4 times 3 and the whole reason I'm writing this way is that now I could write it in terms of factorial I could write this as 5 factorial 5 factorial over 2 factorial over 2 factorial but then you might have the question well where did this to come from well you know I have 3 seats where did this to come from well think about it I multiplied 5 times 4 times 3 I kept going until I had that many seats and then I didn't do the remainder so the number of so the things that I left out the things that I left out that was essentially the number of people minus the number of chairs so I was trying to put five things in three places so five minus three that gave me two left over so I could write it like this I could write it as five I me were doing use those same colors I could write it as five factorial over over five minus three which of course is two five minus three factorial and so another way of thinking about it if we wanted to generalize is if you're trying to put if you're trying to figure out the number of permutations and there's a bunch of notations for writing this if you're trying to figure out the number of put permutations where you could put n people in our seats or the number of permutations we could put n people in our seats and there's other notations as well well this is just going to be n factorial over n minus R factorial here n was 5 r was 3 and minus 5 minus 3 is 2 now you'll see this in a probability or statistics class and and you know people might memorize this thing it seems like this kind of daunting thing I'll just tell you right now the whole reason why I just showed this to you is so that you could connect it with what you might see in your textbook or what you might see in a class or when you see this type of formula you see that it's not you know coming out of you know it's not some type of voodoo magic but I will tell you that for me personally I never use this formula I always reason it through because if you just memorize the formula I was only wait that does this formula apply there what's N what's R but if your reason it through it comes out of straight logic you don't have to memorize anything you don't feel like you're just memorizing without understanding you're just using your deductive reasoning your logic and that's especially valuable because as we'll see not every scenario is going to fit so cleanly into what we did there might be some tweaks on this we're like you know maybe the only person be like sitting in one of the chairs or who knows what it might be and then your formula is going to be useless so I like reasoning through it like this but I just showed you this so that you could connect it to a formula that you might see in a lecture or in a class