How many permutations are there of the 26 letters of the alphabet in which 5 vowels are in consecutive places?

Video transcript

- [Voiceover] 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 two, 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, we could say, well, we have five different ... 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 times four, we have 20 scenarios where we've seated seat one and seat two, how many people could we now seat in seat three for each of those 20 scenarios? Well, three 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. For each of those, how many possibilities are there for the fifth seat? Well, one. For each of those scenarios, we only have one person who hasn't sat down left, so there's one possibility. 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, let's see, 20 times six, which is equal to 120. We have 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 these five people, but we don't have as many chairs, so not everyone is going to be able to sit down. Let's say that we only have three chairs. We have chair one, we have chair two, and we have chair three. 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? I encourage you to pause the video and think about it. I am assuming you have had your go at it. Let's use the same logic. If we seat them in order, and we might as well, how many different people, if we haven't sat anyone yet, how many different people could sit in seat one? Well, we could have, if no one sat down, we had five different people, five different people could sit in seat one. For each of these scenarios where one person has already sat in seat one, how many people could sit in seat two? In each of these scenarios, if one person has 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've seated seats one and seat two. For each of those 20 scenarios, how many people could sit in seat three? Well, we haven't sat, we haven't seaten or sat three of the people yet, so for each of these 20, 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 my brain, whenever I start to think in terms of permutations, I actually think in these ways. I just literally draw it out because I don't like formulas. I like to actually conceptualize and visualize what I'm doing. But you might say, hey, when we just did five different people in five different chairs, and we cared which seat they sit in, we had this five factorial. Factorial is kind of neat little operation there. How can I relate factorial to what we did just now? 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 to think about what we just did, is we just did five times four times three times two times one, but of course we actually didn't do the two times one, so you could take that and you could divide by two times one. If you did that, this two times one would cancel with that two times one and you'd be left with five times four times three. The whole reason I'm writing this way is that now I can write it in terms of factorial. I could write this as five factorial, five factorial, over two factorial, over two factorial. But then you might have the question where did this two come from? I have three seats. Where did this two come from? Well, think about it. I multiplied five times four times three, I kept going until I had that many seats, and then I didn't do the remainder. 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. Five minus three, that gave me two left over. So I could write it like this. I could write it as five ... Let me use the same colors. I could write it as five factorial over, over five minus three, which of course is two, five minus three factorial. Another way of thinking about it, if we wanted to generalize, is 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 permutations where you could put n people in r seats, or the number of permutations you could put n people in r 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 five, r was three. Five minus three is two. Now, you'll see this in a probability or a statistics class, and 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 some type of voodoo magic. But I will tell that for me, personally, I never use this formula. I always reason it through, because if you just memorize the formula, you're always going, wait, does this formula apply there? What's n? What's r? But if you 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. That's especially valuable because as we'll see, not every scenario's going fit so cleanly into what we did. There might be some tweaks on this, where like maybe only person B likes sitting in one of the chairs, or who knows what it might be? 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 to it a formula that you might see in a lecture or in a class.

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