Problem 37 Easy Difficulty
Video Transcript
I always like thes a counting principal problems where we have four men and four women. Um, and there's just eight seats, so I like to put down that there's eight seats. 12345678 And accounting principle. I guess I should mention, is basically that were, once we establish how many options there are, I always forget if it's a e l l e r a l But anyway, the counting principle is, once you establish, you multiply. So how maney options can we have for the first seat? Eso in part? A. I guess a label are a We want the women seated together and the men seated together. So if basically what it boils down to is, we need four women to be written first on Ben, and I'll talk about that in a second. Um, because I guess I'm under the assumption that the women choose their seats first, and it doesn't have to which I'll get to that in a second. So there's four women that could sit in the first seat, but that woman, once she decides to sit right there, there's now on Lee. Three women for the second seat because the first woman, uh, can being both seats at the same time on that's what I meant. By now we're gonna multiply four times. Three. Well, how many women can be in the next seat? Or there's two women left and there's only one for the last option. So that's assuming that the women are first. But then over here now, any man consider right here. Well, there's four men to choose from, and once one sits down, there's only three left. And there's three options for the next seat. Times two and then there's only one person left. Now here's the thing is some people stop right there. But it doesn't say in the problem that the women have to be seated first. Eso what could have happened is everything could have happened the same exact way. But then flip flop. So what I'm gonna do is multiply whatever that answer is by two, and then the correct answer. Once you do, I'm gonna trust you. Use a calculator four times. Three times Two times one times, four times, three times, two times one times two is 11 52. So there's 1152 ways that that could happen. Which brings me to Part B. It's the same counting principal. My thing fell there, except we're seated alternately by gender eso the way I understand that it is really the same. It's really the same problem because, let's say for one that we I want the man to sit first. So I put man and then woman, but it doesn't say it has to be that way. Um, you could have a woman first and then a man, but anyway, there's four men first. And then, uh, once a man is sitting right there, um, it has to be a woman. So there's four options for the next, um, and then there has to be a man next. Well, now there's only three men. Um, and, uh, I guess what? It just boils down to this four times. Four times, Three times, Three times, Two times, Two times, one times one. Because these last two, it has to be the last man, the last woman. But just like before. What if we switch them well, we'll have the same number of possibilities. We could just multiply by two. So the correct things for Part B is also 1152 ways. As I was doing this, you might have heard me starting Thio like think of a different way to do this. And here's another possibility. So I just wanted to share This is what if you said that any person could sit next in the first seat. Okay, so there's anybody there now. The next one has to be one of the opposite genders, so that would be four. Uh, but then the next one, Uh, well, whoever decided to sit here, they chose the gender right there. So that gender is already sitting. So now there's only three left for that gender. So then the next one would be OK. Well, there's only three left for this gender times two times, Two times one times pretty much. All the work is the same. And you'll notice you'll get the same answer. Eso That might make a little bit more sense than you know. And maybe I didn't explain it well enough that you randomly multiplied by two at the end. But I tried my best. So you get the same answer regardless of how you think of it. 11. 52 for both A and B
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Viji Viji aur abhi first arrange the girls that if we write girls so if we arrange cause then arrange voice so there are how many ways to ease of doing this so now we can also see De that his voice can be arranged in a Taurus 4 ways in their places and girls can also be arranged in factory food places so we right now you can see boys can be arranged in a set of four ways and girls when we arrange in
Purvesh sunao ABCD total number of ways than total of number of ways can we do we write total number of ways so this will be factorial 4 X factorial + factorial 40 taiyar two ways of doing this at them so this will come to X factorial 4 into factorial 4 now UN No 2 x 4 is 24 and this is 2410 it become 576 into two so this will give 1152 so this is our final answer so this request
thank you