time. Today we're gonna talk about combinatorics. You know, kind of counting principle, permutation, combination. And this is my gift for you. This is from 2004, 21 years ago. Wow, about whatever. And question number nine. Let's see who's gonna say which one should be the right answer. Let me start the timer for you guys. All right, more and more friends are joining, that's cool. All right. It is one minute. Let's just wait a little bit more. Since we have only two answers so far. Guys, if you think answer is B or for others as well, can you share some of the ideas as well? If it should be B, why? If it is 20, why? So I believe we don't get any more answers. Am I right? But what an explanations our friend says At least for two friends says be how they got ten. I just wonder all right i believe this is 5 choose 2 is equal to 10 am i right so all right this is how they think all right guys let's start it has been four minutes anyway i believe we're gonna have like more participants we'll see okay as you see in that stand guys the ice cream cream stands there were five different flavors okay we have a group of children we don't know how many of them came to stand and each child bought two scoops of ice cream with two different flavors these are important because it depends with those keywords so you get different answers i underline those for you if none of the children choose the same combination of flavors okay they all get different and every such combination of flavor flavors was chosen how many children were there think about guys we know that one kiddo can choose five different first of all we have one and two scoops as you see for the first scoop any kiddo can choose five different flavors am i right just think about guys i mean there is no restriction for that for the second scoop we know that they can choose since you got one of them and there are five of them you can choose four different scoops since they are separate events you could say five times four is equal to 20 but here is the deal do you think like let's say among those five scoops you have a and among those scoops you have b we know that a b is one of the option here also b a is another option here think about guys do you think a b and b is different or not at all i mean you can have like you know chocolate and strawberry or strawberry and chocolate it doesn't matter which one is at the top of another well in that case we need to eliminate one of these two options in that case you need to divide those by two guys and you should get 10 the question do we have any question about that one the first scoop and second scoop i just show you different options I believe there is nothing so I'm gonna move on okay also I believe some of our friends says you can just find all of those difference like a b a c a d a e and so far if we have like a b c d n all of those different options I am kind of trying to check the chat as well guys if since it's not really too much of there are not too much of huge numbers we have five I would say a b c d e let's see these are the different options so from those we're gonna choose to you know I can have a B start with a that I can have a C hope look at my great P by the way I have a D and I have a E since I already choose a even I start with BA it's gonna be same thing I'm not gonna choose again I'm gonna start with B I can have BC BD and be as you see okay this is supposed to be and I can have CD and CE and lastly we can have the well if you can't all of them guys you will also see 10 different options okay I hope it makes sense no one says anything that means it makes sense I really wanna believe that all right let's move on as you guys we are in the lesson 8 today I am gonna remind you how to solve problems underline them make sure you understand make sure answer makes sense let's just go over some examples today the fundamental of counting principle states that when there are M different ways to do one thing and N different ways to do another that means then there are M times N ways of doing both for example how many outcomes are there when answering three true or false questions well there are two ways either true or false to answer first question we won't know that followed by two ways to answer the second one and also two ways to answer the third one so the total number of outcomes should be 2x2x2 equals 8 also we can use the least math here if you want again guys if numbers are not big you can just use this method as well I just show you for the warm-up question as well here is how it works but to be able to make sure you understood what I'm gonna change a little bit numbers then I'm gonna challenge you then you will give us the answer liquid instead of how many outcomes are there when answering three true false questions what if we had six questions this is your challenge guys look here is example this explanation I only change a number please find the answer whatever you get and share with us you can guess to no worries There's someone who says 2 to the power of 6. What's the answer? What's that number then? 64. Okay. Anyone else? So do I have only one answer so far? Yeah, only one answer. That means you don't get it, guys? Look, we have definitely more than one kiddos here. If anyone else doesn't know the answer... Remember, guys, we only multiply those numbers here because these are not dependent events. Whatever you answer for the first question doesn't affect your answer for next question, or other questions, or so forth. That's the reason we can just multiply them, you know? Anyone else for this one? If we change the number, how many outcomes are there when answering six true or false questions? So I'll look like no one and no one share anything with you. If this is the case, guys, I'm going to just show you and move on. Here we can just use the same logic, you know. If we have six different questions and if we have only two options to answer each of them, which is either true or false. We have two different options to answer the first one. With the same logic, two for second, and so far. Four, five, six. We can say 2 to the 6th power. Thanks to one of our friends who said that. Or you can find 2 to the 6th power is equal to 64, guys. Think about when you have eight outcomes, when you're answering three questions, but I double the number, it doesn't necessarily mean that you are going to have double the outcomes, which is 16. It's not the case. I gave you number six on purpose. Hopefully it makes sense. Another example, how many ways are there to line up four people? Well, we can make four spots and find a number of ways to fill them, you know. Unlike the true or false question in example one, after a person is choosing, well, he or she is not available for another spot. There are four people to choose from. First one, then only three left for the second, two left for the third, and the last one for the last. So the answer would be four times, three times, two times, one, which is 24. This question is designed to also remind you, I mean, teach you the new term factorial donated by that little symbol is the product of all positive integers less than or equal to a given positive number. For example, if you have five factorial, that means that that term is equal to one times, two times, three times, four times, five, which is 120. Zero factorial defined as one and many combinatorics problems can be answered by using factorials like example two. Okay, so I want to ask you guys if you get it or not. To be able to make sure you get it, I am going to ask you, instead of lining up four people, what if I ask you to line up six people? What would you do? Please find the answer and share with us. It is literally the same logic, guys. Let me give you some time. If you feel you get it, you need to prove to me, guys. okay so this is the what they shared with us or you find them sharing there's their chat messages quoted okay all right thanks so much so in that case i believe we have one friend give us one answer what about other friends guys by the way it was 45 seconds two different people okay two different people same same as same as that's good okay we are getting there cool come on guys it's not that difficult look i only change the number that's it not even change the scenario okay it has been one minute five seconds so remember we just talked about the factorial uh term guys for those cases you can just use six factorial let me just put it here which is equal to one times two times three times four times five times six gives us 720 guys i hope that makes sense if it's not please ask us all right i'm gonna move on you all right permutation is the number of ways things can be arranged or chosen where the order matters let's just go over the example you have five pictures and you want to choose one to hang in the living room and one in the kitchen how many ways can you hang the pictures well we can assume that the first one chosen is for the living room okay and the second one is for the kitchen in that case a b and b a represents different ways the order matters we know that there are five pictures to choose for the living room okay and we get four left for the kitchen so we should have five times four twenty this is also the same as finding factorial of five divided by five minus two factorial because we choose or order two of them for that case okay in general the permutation of n objects taking r at a time represents that symbol represents n permutation r it's here the formula okay after we check this out i'm not gonna just read this one like a reading class this is a math class guys that means you gotta learn by practice what if you had seven pictures and you want to choose one to hang in the living room and one in the kitchen how many ways can you hang the pictures guys Okay, I'll give you some time, maybe one minute. All right, it has been one minute and seven seconds. I believe you get some answers Let's see Okay, we get 42 because it's time seven there are some pictures for living room and six for the kitchen Okay, we get 42 from another friend as well. Thanks so much guys for sharing your ideas And yeah, we should just have Seven of other choose seven times six or In that case you can say seven factorial over seven lines to You get seven times six times five factorial over Five factorial they cancel each other. So we should have seven times six case. Actually, I should just put it here. It's a little visible It stops to like You get seven times six times five factorial over five factorial then you get 42 All right, any question for this case? What about combination? Is the number of ways things can be chosen where the order doesn't matter? For example, there are 20 students in the class. The teacher wants to choose three kiddos to move the chairs How many different groups could the teacher make? Well Let's say the students ABC are choosing the way Can be they can be chosen by the teacher in different orders like ABC BACC B Whatever with the total of three Factorial ways, but these all result in the same group in that case. We can say that order doesn't matter We do it like a permutation But we need to also divide by three factorial because all of those represents the same thing basically In general combination of n objects taken are at the time represents with this symbol Which is n factorial or n minus r factorial times r factorial Okay, you know I am gonna change a little bit. Let's see there were Let me think about instead of tennis shoes. I'm gonna give you easier one There are only ten students this is kind of small group and teacher wants to choose Let's just make again three, okay, so I'm wants to choose three. Alright, we can keep the three then I was gonna make four but it would be so easy. Okay, just make it three again. Alright, please Well Take your choose three kiddos to move the chairs How many different groups could the teacher make guys out of ten kiddos instead of three? I mean instead of 20 kiddos, okay Let's see what are gonna find and please share your answer just start the timer It should be one of the easy integers, yeah. It has been 24 seconds by the way. 38. 50 seconds. Looks like some of our friends find the answer. It shows me. Okay, one of our friends find the answer, which is 120. What about others? Only one friend? Come on, man. So I assume we have only one answer. Am I right? Guys, we definitely have more than one kiddo here. We have a lot more than one. What about others? The only thing you got to play with the numbers. Like, let's assume you have no idea how to solve that. You have given the answer. Here, you just work with little different numbers. That's it. It's the only one answer. Even you say nothing, I assume that you don't get it. It's tough, man. Which part you don't get it? Can you share with us? Look, I heard from many, many kiddos. Combinatorics might be a little confusing. That's the reason I'm just using a little more time here to show you. I shouldn't get only one answer, man. Others, are you sleeping or what? Are you still in the break mall? I need at least one more answer, guys. We'll get the right answer for this one. I don't see any other answers in the chat. That's sad. I wish we had an option to, you know, ask them to put the camera on. You'd see what other friends are doing. Hopefully, you're not sleeping in front of the camera. Okay, guys, I am messing with you, but remember, we use just combination rule here. We had 20 kiddos. The rule says 20 factorial divided by 20 minus 3 factorial times 3 factorial. Because order here doesn't matter, as you see. A, B, C, B, A, C, C, B, A, or whatever, they all represent the same group of kiddos anyway. So we just apply the formula here. It's the same logic if you had like 10 kiddos. Oops, someone shared something with the chat. What's that? Okay, another friend also find the same thing. Thank you so much, guys. Right now, I can say that at least one more of your friends get it. So here, guys, we can say if we change the number to 10, we have 10 factorial over 10 minus 3 factorial times 3 factorial. Look here, we know that 10 minus 3 factorial means 7 factorial. I'm going to make it 10 times 9 times 8 times 7 factorial over 7 factorial. I don't have to put those numbers next to each other to multiply, whatever. If you see something common, just cancel those, and we get 3 factorial here. Well, we get 10 times 9 times 8. I can continue from here, I believe. 10 times 9 times 8. You see, I didn't multiply them yet. It's not necessary. We get over 3 factorial, which is 3 times 2. I also keep 3 as a 3 times 2. Well, we have 9 at the top. There is only multiplication here. Just divide to get 3. We have 8 at the top and 2 at the bottom. Simplify to get 4. We get 3 times 4 is 12 times 10 is 120. I hope it helps. Kind of trying to give you some shortcut ways because it's going to be time test, guys. So, you should be able to do that. Okay, I believe we are good. I really believe that. And the next one. Some common strategies use symmetrical nature of the problem to count. Example, if all of the members can be chosen in the same way, then you can find out the number of ways of choosing any one member. And then multiply by the number of members. For some cases, you can use case work if the nature of different members are different. One example, how many squares of any size are in a 3 times 3 grid? You see, we just try to show you. If you are trying to find one by one with that size, we have 9 of those as you can count. If you are looking for 2 by 2, we are going to have 1, 2, 3, and 4 of those. And if you are looking for 3 by 3, we are going to have only one of them. At the end, you will just add them. 9 plus 4 plus 1 is giving you 14. Sometimes, we can use complementary counting. Instead of counting the favorable outcomes, you count the rest. Then subtract it from the total. For example, if you flip a coin 5 times, in how many ways can you get at least one head? You can count the total of every single outcome. Count how many ways you can get no head. Then subtract from the total number of ways of flipping the coin. So, the total should be, since we have 5 coins, or a coin flipped 5 times, we have 2 times, 2 times, 2 times, 2 times, which is 32. We can only get one way to get no heads. Then you subtract to get 31 heads. I hope it makes sense. Is there any question? I really want to move because we got to discuss some questions here. It looks like no one says anything. I'm going to move that. Okay. All right. This is my first gift for you from 2011. And question number 8. Have fun, guys. Let's start the timer for you. We have 23 seconds. 45 seconds Oops no answer. Okay, I'm gonna give you more time then guys All four-digit numbers, the sum of those digits is four. I would think first which digits I can use, you know. And we all know that four-digit number cannot begin with a zero. That means the smallest four-digit that I have, I should start with one, you know. Okay, oh, two people says, oh, three people says no, okay. All right, we get some answers, that's good. Since we get some answers, unfortunately, I would like to say that that's close but not the right answer yet. I wonder here right now, any of friends says B, they might want to move to another answer choice. This is the time teacher feels like a failure. Come on man, three kudos and they all get the wrong one? Again, I am saying this is your chance to switch your answer to something else. Or I am going to start this all anyway. Look, let me give you a hint. You can start with a smaller number, one, okay, someone says 15, that's good. Unfortunately, it's not right either. Come on, look at us. Okay, so the smallest number, guys, should be one, okay, someone says C, C is good. After one, guys, you can put some zeros. We can put as much as zeros here to get smaller number because we get a smaller number, you know, because they said we arrange in order from least to greatest, as you see, okay. From smaller, I can put one more zero if it happens, if it's fine. Some of digits is four, then I can just put here three, guys, as you see. In that case, we all know that the smallest number should be 10, 3. Well, what about next number? Still, I can keep this one zero, you know, the hundreds digits. I have one and zero. Do you think I can change those numbers here? Can I just make it one and two here? What do you think? Oh, you cannot answer, that's sad. I thought you can answer me. Then with this logic, we can keep going. Am I right, guys? Because after 10 or 3, I would just go with 10, 30, but well, maybe we have smaller numbers. As you see, that works too. After this one, I will only change, I mean, switch one and two, you know, I can get 10, 21, as you see. What else? Then we won't know, we can still use that three to make it 10, 30, okay? What else? As you see, after that time, guys, well, 100 digits was always zero. I can make that 100 digits one. Okay, we start with 1,000, then 100, then I can make the smallest number, which is all two, as you see. I can still make it 11, then make it another 11, you know. Then I can just switch those zero and two, 11, 20, right? What else? Let's see. I may not start with 100. Maybe I can make it 200 after 1,000. I have 1,000 and 200, then zero, one, you know, basically. Then I can make it 12 and 10. I cannot make it zero, two, you know. What else? Think about it. After 12, I can make it 1,300, you know. 1,300. What else? I cannot make more than that, so then I should continue with 2,000, guys, after this one, as you see. And when I'm talking with the 2,000, the smallest number should be 2,011. All right. In which position is the number this one? Let's just count. One, two, three, four, five, six, seven, eight, nine, 10, 11. That means I just missed one of them. Let's see. Suhan, which number I missed, man? I missed something. You have to put as many zeros as possible after the two. So 2,000, two. Oh, you're right. After this one, I should make it 2,002. You're right, man. I missed that number. Yeah. I just jumped on the solution. OK. So after two, we have 2,002. Then can I make it 2,020? About 2,020 is greater than 2,011. So then it's going to be 2,011. OK. That's one. So we're done. Six, seven, eight, nine, 10, 11, 12. As you see, it should be the 12th number. I really assume that those two friends who says nine already moved to C, answer choice C. I really want to believe that. Or any question about this one, guys? No? OK. I believe that you would just put something on the chat. OK, next. All right. Number two, which is from 1998. And question number 13. Okay, one friend says the answer should be D, what about others? Do I have only one answer? No, no way! okay it has been 10 minutes or that's good I mean not 10 minutes I meant two minutes okay so four friends says eight guys think about we have three married couples in how many ways can you form a three-person group look for first person in the group can I just choose one from two since we have those three married couples for the first person I can either choose husband or wife you know this is the first couple but I am separating them and I choose one on the one people want to do one person whatever out of two it's same logic we have second married couple it's same logic we have the last married couple I can multiply them because I am choosing I'm able to choose them without any restrictions you know it doesn't affect each other they are sprayed events then you should get eight okay or if you wanna you know use this method diagram method whatever method you can use that okay number two Let's start timer for you for this one as well. Okay, interestingly, we get two different answers. Well, well, well. If I say both of them are correct, are you gonna cry? I mean, both of them are wrong, are you gonna cry? But I will. It's neither A nor B, guys. Let me check your answer one more time. So what's going on today? Okay, someone changed their answer, that's good. Well, okay. Let me make E, then, yay! One of us has to be right, there's no other case. I said answer should be E. Okay, guys, answer was not E, but answer was not A or B either. So please think about between C and B. By the way, it's two minutes already. Only two friends says D, I guess. All right. So we have gained six points, guys. As you see, we're select on the parallel lines A and B, four on the line A and two on line B. Well, well, well. If we didn't have any restrictions, what would you do? Would you just choose three points out of six? You know, because we need to, if we had those points on the same line, or if they were not straight, anyway, we would choose three random points out of six, you know. But right now, four of them on line A, but what does it mean? If I have line like that, one, two, three, four, can I make any triangle? Is it possible, guys? I assume it's not. Okay. Let's see, we have four points on line A and two points on line B, okay? What can we do here, guys? If I start only one vertices here on point, on line A, we should have one, you know, like that. Oh, we have more answer. One, two, three, and four. Does it make sense? We can get from four, if the vertices here. Or, if we choose one of vertices here only on the one point here, we can get one triangle here like that, you know, or two triangle, or three triangles. Am I right? Do we get more, Soham? Of course we should get more. You may say, we get one, two, three, or maybe this one with four. You know, I would get all different types. I'm just trying to show with vision. So we should get four. Am I right? More? More than four. Let's see. One. Oh, you're saying we can get all of those separated ones. One, two, three, you know, with only, I get your point. But, Soham, here is the deal. We already count those with the, oh, no, we only count from top. Yeah, these are the four. Okay. If we kind of, you know, make them upside down, we can get one triangle. You're right, buddy. The second one, this is what you meant, yes, Soham? It's the third. Am I right? Then you gotta keep counting. This is three, and this is four, and this is five, you know, and the whole is six. It should be right, man. With the same logic, we're going to get like 6 more from here, then it's going to be 16. Look. Let me change the color, what if I just choose kind of a green color. So with that point, this is going to be an upside-down triangle, we have 1 here guys, 2 here, 3 here, then we just make it double y-base, 4 here, 5 here, and the whole shape is here, which is 6 as well. It should give you 16. But the question is, you're going to say, but sir, what if we have like more and more points? Exactly. This is what I've been making. How can we use with the formulas? Okay. What do you think about that part? So guys, you get 16. Did you use the method I just showed you, or you just used the actual method? Okay, someone shared their answer, I believe. It was actually me. Oh, okay. So they didn't share their answer, we can wait for them to share their answer as well. Yeah, let's wait. Okay. All right. that makes sense so you are saying those are line a and line b or a times b as a you know base times height as a train i believe you are making line a and line b am i right so on those are the number of points on line a and line b okay that makes sense what about our friends guys i don't know if this is anything All right, guys, look, we had six points, am I right? We get six, oops, it doesn't look right. Six combination of three, am I right? If we had those random points, basically. But, there is a case, since line A has four of them, it's not going to be, you know, form a triangle from four of those points in the same line. You gotta take out four combination of three. Then, you can solve it. This one gives us, guys, six times, I mean, six factorial, basically, over six minus three factorial, times three factorial, minus four factorial, over four minus three factorial, times three factorial. So, what we should get here, guys, is six times, five times, four times three factorial, over six minus three give us three factorial, times three factorial. As you see, we can cancel those. Yes, I just write it down, expand like that. Four factorial means four times three factorial, over, this is four minus three, just one factorial, just basically one. We get three factorial. They cancel each other, guys. We have four left over here, and here we get six times five times four, divided by three factorial. We both know that three factorial means three times two times one, which is six. This one is also canceled. We have five times four is, let me just put here, 20 minus just four, also 16, guys, okay. Look, I don't really mind. Whichever method is easier to you, just use it. I try to show you the visual one, because for some kudos, that makes more sense. But like I did, there's a big chance you may do some calculation mistakes. Just make sure you count every single shapes or objects. Any question for this one, guys? I believe there is no question. I'm gonna move on then. All right. So, guys, next one. Number four. All right, one friend says it should be 15, okay. What about other friend? It has been 1 minute and 8 seconds so far. Should I say this is not the right answer? Are you going to be mad at me? But this time I really want to say that this is the right answer. it has been 1 minute and 40 seconds guys i hope we will get one more answer from some friends It has been two seconds, I mean, two minutes, guys. I wonder if that friend who says answer is D can share the answer or something with us. So how much do you, hopefully not. Wait guys, let me give you a hint. You know that that calculator is broken and it doesn't display the digit one. In that case, at the end, if we get 2,007, can I guys assume that display should be something like that, you know, because we might have some of those missing ones here. Think about that, okay? We both know that Mike typed, guys, whoever says 16, buddy, I have been telling you, D was the right answer. You were just checking for others, I believe. If people are sleeping or not, okay, that's a good try. So look, as you see, Mike typed a six different, no, six digits, number into that calculator, okay? In that case, since 2,007 has four digits, we are missing or we need two more ones. Look, there are two different case, guys. Remember, we have been talking about case studies. If those ones are together, I can put those ones here or here, here, here, or here. We have five spots. For only those two one digit spots, they are together. So I can assume that they are together. In that case, I can say five combination of one gives us basically five, am I right? You know, if you just do the calculations, I believe you get that. Or second case, let me just say, case one, they are together. Case two, ones are separated. Look at my great handwriting, we cannot read that, oh boy. Okay, separated, you can still not read, hopefully you will one day. Okay, if those ones are separated, we have one, two, three, four, five spots to fill in two numbers, basically. Either they are same or different. I mean, since they are same, you gotta use this one. Then when you calculate the combination, you gotta get 10, guys. It's time, that's the reason I'm kinda making in a hurry. We have 10 cases here, plus five cases here. When you add those cases, guys, you should get 15. Any question for this one, kiddos? What about alternate solution? Yes, let me erase that. You can just share with us one, you can annotate. Yes, but I would just erase everything, just show us, please. Oh, for the alternate solution, so you have these six-digit number, and you know that 2007 is what remains. So there are two digits of one, which the ones you can place in six choose two. Six choose two. That's 15 ways to place the ones. And then, so suppose there's a one here and a one here, then the remaining digits have to be two, zero, zero, seven, and in that order. So there's only one way to replace the remaining digits. So the answer is 15. Yeah, that makes sense. Thanks, man. All right, kiddos, I hope it was helpful. And you said there is no question. Then I'm going to see you next time, guys, okay? Take care, bye.

combinatorics

permutations

combinations

counting principle

factorials

mathematical principles

groupings

arrangements

problem-solving

outcomes