 Oops quick return half an hour break back to math max another live stream So yes a nice conversation on the last live stream, which we finished about half an hour ago. So we're back on it again Another just open discussion math-centric, I guess live stream for those that Need a little math out towards the end of the year school year if there's any questions and stuff There's some quite interesting questions that came on last on the last stream just some randoms just brain teasers and a couple of combatorics questions and Thank you for Was that was Asking the questions. I was fantastic. There's a couple of people that were replying answering the questions really fast Which was fantastic, and we ended up doing them on the board here, and this is the To do the problems if there's any questions that coming up Okay, so we'll just keep this going and I'll be here again tomorrow for anyone that Requires any math help If you want to Run some stuff past me as long as I can do them. There's one question that stopped me in the last stream I forget what it was. It was What was it? I Forgot what it was and one of them was a visual one, which was fantastic, you know a pond if the lilies and a pond double Every day and it takes 48 days to fill the pond How many days does it take to For the pond to be half full of lilies, right? And the answer is one day less than it takes for it to be a full pond of lilies We'll just chill. I could do a lesson some of the stuff I've been reviewing with some of my students, but we'll leave that be for now until We get people popping in Drink some tea case it was here in the last stream and you need to take a break Taking a tea break if you're feeling low The weather is weird right now warms up cools down So I know some people that are having a hard time with that allergies kicking in and they're getting a little bit of aches and pains with the weather adjustment We're like it's like metal right our bodies when it gets cold With metal it shrinks a little bit when it gets hot it expands a little bit. I think our bodies do the same thing They're pretty sure we did a binomial Check this out because I was just working with a student yesterday. We're doing our class couple of days We're doing some binomial Distribution, I don't know if we did that on a stream previous stream or not quadratic functions Ferris wheel I don't think we did binomial distribution Rational functions graph It's a fun thing to do with the binomial distribution because combinatoric permutations came up a couple of combinatorics permutations factorials In the last stream, so let's do you know what we're waiting for people. Let me just do a binomial Distribution Or Pascal's triangle, that's what I want Pascal's triangle This binomial is what Pascal's triangle. That's what we want Yeah, let's do Pascal's triangle Okay Put this down and let me change the Change the screen and we'll talk about we'll formalize Pascal's just so you know we'll formalize Pascal's triangle at some point and get into probability statistics Because Pascal's triangle goes into a lot of different things But let's do this So one thing we have in mathematics is we expand And foil out different types of binomials, right? So for example, if we have this X plus Y Let's say Let's do let's take it down from the beginning, right? X plus Y X plus Y to the power of zero will do X plus Y to the power of one We'll do X plus Y to the power of two We'll do X plus Y to the power of three and we'll do X plus dot dot dot X plus Y to the power of seven Okay Now make sure this is working out. Okay. Yeah So take a look at us Let's see we had a binomial expression a binomial is just basically two terms, right? So we want X plus Y to the power of zero Anything to the power of zero is one. So this one is just easy one Okay We want X plus Y to the power of one Now anything to the power of one is itself. So this just becomes X plus one And then we want to expand X plus Y to the power of two now X plus Y to the power of two Squared means whatever is in there whatever's being squared times itself, right? So this is going to be X plus Y Times X plus Y and then we foil us out. We've talked a lot about this, right? So this becomes X squared plus X Y plus X Y Plus Y squared and that simplifies down to X squared plus two X Y plus Y squared Okay So I'm just going to write that whole thing there. We don't need to work so that becomes X squared plus two X Y plus Y squared Okay Let's say we want to expand this now. Let me erase this one. That's to the power of seven So this thing says X plus Y times X plus Y times X plus Y now the way we have to do this is we've got to do two of them and then Combine like terms and then connect that up with the third one, right? So we expand this guy and if we expand this guy, that's what we end up getting, right? So this becomes X squared plus two X Y Plus Y squared Plus Y squared times X plus Y Okay Now what we're going to do is we're going to expand this this times this this times this and this times this and so on and so forth Okay, so this becomes I'm going to move it over here because we're going to need the space So X squared times X is X cubed X squared times X Y is plus X squared Y Two X times X is going to be plus two X squared two X. Oops forgot the Y part Two X Y times X is two X squared Y Two X Y times Y is plus two X Y squared Y squared times X is plus X Y squared Y squared times Y is plus Y cubed And then we're going to combine our life terms So we have X squared and X squared X squared Y and X squared Y these guys combine and we've got XY squared and XY squared this guy and this guy combined. So this becomes X cubed plus three X squared Y plus Three X Y squared plus Y cubed Okay that is This guy Expand it Now take a look at the amount of work required to go from here to here. That was ridiculously easy This was just writing this out. This was foiling that out. It was fairly quick, right? This guy took us much longer because the first part of it was us expanding this here But then we had to multiply by this guy. So it takes a lot of effort to get to that point, right? Now just imagine trying to do this for let's make sure this fits on the board here X plus Y to the power of seven Right the amount of work required to do this is going to be huge right But in mathematics, this is just a pattern of things in mathematics. What happens is We've taken a look at these things and we've come up with Sort of shortcuts simpler way to do these things, right? And that's what we're going to talk about right now Now just imagine if we wanted to expand this and let's before we get into this Let's see what the pattern is here. Okay, so that's going to be three X Q plus three X squared Y Y blah blah blah blah. So I'm going to copy this here. Let me just erase this Okay, that's the expansion here. This is X plus Y Q is going to be X Q plus three X squared Y plus three X Y squared plus Y squared Y cubed Okay, which is should be what we have here, right? Okay, we'll keep this down here. We'll expand that now Looking at this and looking at more expressions more expansions of these Binomial expansion yes binomial expansion If we look at this and we continue this let's assume we've continued this to X to the power four X to the power five So on and so forth What we end up getting is this The binomial expansion does this. Let's see if this is going to be We write the one here, okay, this is going to be hard to erase but that's okay Over here, we're going to assume. There's a one in front here and one in front here So we're going to expand this one one In front of any variable that doesn't have a number is a one. Okay, so there's a one here and a one here So right now all all all we're looking at is the coefficient In front of the variable. So there's a one there. There's three there three there And a one there right and a two here, right? So we have one one one one two one One two one One three three one one three three one right There should be a pattern there that you're seeing and the way this works is this The ends are ones just going down like this And these numbers here are these two numbers added up So whatever number you're putting here, you're adding up the two number that is flanked by so that's One plus two is three Two plus three is three. Okay. That's one pattern. We're going to look at for the binomial The other pattern we're going to see is this This is x and that's y right now that's x to the power of one y to the power of one Okay Same here same here same there right Now what you're going to notice is this If that's to the power of one is just whatever they were if that's to the power of two What happens is the power of two is here on the first one And then for this is the power of one And then for this guy the y we start off with the power of one And we're ending up y to the power of two Over here is to the power of three, right? We're expanding a binomial to the power of three the first variable Gets to the power of three the last variable is to the power of three In the middle what happens is The first variable's power Decreases by one increment at a time. So that that's x to the power of three x to the power of two x to the power of one And x to the power of zero but x to the power of zero is just one So we don't expand we don't bother writing it down, right? And over here the y We have y to the power of three y to the power of two y and over here we really have Y to the power of zero So there's another pattern for the variables for the terms here in the binomial Expansion, okay, and that's something we're going to use as well We're also going to keep a note on this and I'm going to lay all this out in one thing We're going to erase this and we're going to lay it all out, right? Here's the other thing you can you're going to notice The first One here is to the power of zero Okay, this is the guy The second row here is to the power of one The third row here is to the power of three The fourth row here is sorry the third row here is to the power of two The fourth row here is to the power uh three Okay Here's another pattern embedded within the binomial Distribution of the binomial Pascal's triangle, right? Is This is to the power of three power of three has one two three Four terms when it's expanded one two three four Power of two is going to have three terms Power of one is going to have two terms power of zero is going to have one term Okay So the way it works is whatever the power is here Okay, you're going to have one extra term than what the power is Okay Since that's the case what we're going to do we're going to call this position zero position one position two position three Okay, it's just terminology The way we're going to express it and that comes out in the combinatorics aspect of things. Okay so Knowing this pattern i'm going to erase all this and we're going to just going to generate Pascal's triangle And relate that to the binomial theorem and expand x plus y to the power of seven Okay, and then we're going to do a little bit more complicated as well Yeah, you get the coefficients for Pascal. I'm just going to read a couple of comments that were posted Does this link to Pascal's triangle? This is Pascal's triangle for sure Pascal's triangle and the binomial theorem binomial expansion And combinatorics permutation. Well, not permutation but combinatorics They they're all linked together, right? So let's erase this Okay Let's create Pascal's triangle Pascal's triangle does this Start off with one And then i'm going to write this tight so we can get a fair bit of Nice expansion, right? one one One two one one three three one Let's generate the next one the end here is one one plus three is four three plus three is six four one And you can see the symmetry down the middle from this side to that side, right? And then one again, this is five one plus four is five Four plus six is ten and again ten five one Expand again one six five plus one is six Five plus ten is 15 ten plus ten is 20 and again repeats 15 six and one This is one seven one plus six is seven Six plus 15 is 21 15 plus 20 is 35 15 plus 20 again 35 21 So that's to the power of zero one two three four five six seven that's to the power of seven That's what we wanted to expand, right? So let's expand this we had x plus y to the power of seven Now if we want to expand this The numbers in front here were just a coefficients Right, so if this is to the power of seven how many terms are we going to have in the expansion? We're going to have eight terms in the expansion, right? Let's check it out one two three four five six seven eight Right, so these numbers Are the coefficients the numbers in front of this expansion. So I'm going to write those down for now So we got one I'm going to leave space for the letters one plus seven plus 21 I'm going to go down this way, right? plus 35 plus 35 plus 21 plus seven plus one okay What we also saw is we start off with the first Variable here first position here and that's to the power of seven So this is going to be x to the power of seven the second guy here The y Is to the power of zero, right? But we're not going to well, let's write it down y to the power of zero. Hopefully this is coming up big enough It is it's pretty small Is one i'm just going to read a couple more comments is Here let's change the look here Let me read these comments is one plus nx plus n That Effectively Pascal's triangle. We're just an experiment. We're gonna we're gonna talk about that We're we're gonna take this and then we're gonna Do the binomial expansion? So we'll cover that but to answer Uh, let me make sure you got all correct I believe so, uh, but we'll do it. We can check it with what you wrote down Yeah, the coefficients of are the numbers in the nth row of the triangle Cool That's without using c I take it. Yeah Or the n with the row. Yeah, perfect. So everyone's been taking care of Thanks racer racer kill answering the questions Okay, so take a look at this So these are the coefficients now we're putting in the variables, right So what's gonna happen is this one here is going to have an x and a y as well x y And the way it works is the power to the first one is going to be decreasing one at a time And the power for the second one is going to be increasing one at a time Remember this position was zero one two three four five six seven, right? That's where the zero comes in here and this zero plays out when we write it in terms of We're just jumping ahead a bit a little bit. We'll explain this This zero comes in seven to the power of zero. That's what we're starting with here To answer Collins's question, right? So we're going I'm going to put an x and a y for all of these for now And then we're going to put the powers on Okay x y x y x y So the x starts off with power seven And then it decreases one at a time. So this is power of six power of five power of four power of three power of two Power of one power of zero, which is really one. We don't have to write it, but I'm doing just to be consistent And then the y is increasing one at a time. So it starts off with zero one two three four five six seven We just expanded x plus y to the power of seven using Pascal's triangle and pattern recognition In a fraction of the time that it took to expand something to the power of three. Remember these guys are Let's write these down here. This is power Power of zero one two three four five six seven right We did this expansion for power of seven In let's say about the same time a little bit less than what it took to do this And to do this By expanding and multiplying together it would have taken an exponentially longer time Right, but we're able to do it by pattern recognition Now because there is a pattern here we can formalize it right we can Take this thing and say, okay, we don't want to generate a Pascal's triangle What if we wanted to expand something that was more complicated than this? Well, let's just say it was this to the power of 21 Right now, we're not going to draw Pascal's triangle into 21 rows or 22 rows, right? Because that's row one. That's row eight, but it's power of seven We're not going to draw Pascal's triangle out large. What if this was 210? Wow, it would be huge, right? So we need to apply some kind of formula to this whole thing And the formula is this Okay, and it you can use it to expand any binomial to a certain power Okay The formula for each term There's one term two terms three terms four terms five terms six terms seven terms eight terms corresponding to This this this this this this this this And we start out with a count of zero right So for this one Right this expansion one other thing I should mention with this as well the powers on The terms here Add up to seven seven plus zero is seven six plus one seven five plus two seven four plus three is seven Three plus four is seven two plus five is seven one plus six is seven zero plus seven is seven, right? So there's a lot of pattern embedded here For each term here. This is the formula we're going to use and I for remember correctly Is going to be c the power we're going to call n. So let me write this in general term Let's say we wanted to expand this x plus y to the power of n For this if you're going to go to any specific term Right, and let's call these guys the position here k No, I'm not going to put it there because I don't want to confuse it with the power Right, we're going to call these guys k's Okay, these guys here this is position zero That's too messy. Let's erase that is make it neater k zero, okay so This is the expansion you have combinatorics choose n This is position zero. So we're going to call this k I believe it's minus one If you want the first position Then it's one minus one is zero, right? And then we're going to have the first term And the second term in there x and y and the power on the first term is going to be n times k minus one and the power on the second term is going to be k minus one Okay, I wrote this too tight. I can't fit it all in You know, let me erase this a little bit more. Let's move this over a little bit That's going to be n times k minus one And that's why to the power of k minus one If I have that correct Oops My bad. That's not n minus k. It's n minus k Geez almost messed that one up, right? So before we continue on this, I'm just going to read some comments just to make sure we're on the right track I didn't mess up any of the formulas here, right? So let me change the view again do you have They have every degree in that no, I don't have I just have like for me I I have my degree in geophysics and a minor in mathematics I don't consider myself to be a mathematician. I just Learn mathematics to a level that I can use it in my life. That's it okay And it didn't I guess it took some effort learning is not supposed to be easy If it wasn't just be doing it wouldn't be learning It took a little bit of effort But thanks to all the questions that my students were giving me I was directed down the right path because I was trying to seek out the answers to the questions that they had So I learned in the process, right? And I really didn't learn mathematics until I started teaching mathematics I just I was a monkey doing what monkey saw monkey does, right? I just learned certain things and I used those things in my life, but I really didn't have an understanding how everything expanded To include mathematics Okay, I believe it's just combinations. Yeah, these combinations. Yeah, the coefficients are also equal to and choose k k Yeah, I'm using k it's n choose k but I'm using k minus one because I wanted to correspond with The position of where they are in the expansion I believe that's the crew. I don't 100% sure if that's right terminology for it, but I think x should be to the power of n minus one and y should be to become two n sum k k k Over k from zero to n. Yeah, it depends where you start the count, right? If you say k the count It starts off at zero, then you don't need the negative one Minus one. I should maybe type this out. See what the correct terminology is for Where is the formula? Where is the formula? Oh, they do do it that way Let's check it out k Okay, so let's do it this way So since that's the case Then we're going to do it this way as well. Yeah, we'll follow the Convention I guess so Let's do it this way So let's do this. Let's call this k and that's I'm going to call this k Okay, that's a wacko k Okay, and the k is the count Right, I should have actually used it from the beginning because I said k is equal to this, right? So the first position is going to be zero here if we want the first term. Okay So let's Use this expansion actually let's use this expansion to find the middle term right off the bat, right So let's assume For this expansion here that we have What if I wanted you to find The fifth term In this expansion and that's sort of a question that does come up a test that they ask you or they ask you to find the middle term and because Power of seven has eight terms in it. There is no middle term, right? And even power will have a middle term because an even power has one more Then the power says one more term than the power says, right? Which makes it odd So there is a middle term. So the power of six has one two three four five six seven terms So the middle term would be 20 and an odd power has eight terms. So there is no middle term The middle term would be between these two guys, right? So let's say I wanted you to find the fifth term in this expansion, right? So what you would do you would go, okay The fifth term power of seven has Eight terms in it, right? So the fifth term would be K equals zero k equals one two three four So whatever term you're looking for This number here is one less than What the term is you're looking for So if that's the case, let me see. Are we showing up here? Yeah, we're going to show here So if I want The fifth term this is what I'm going to write Um, I don't want to erase all that. I'm going to take this formula and put it up here. Okay, so that's going to be c and choose k right and then We're going to call it x for now x n minus k And y to the power of k Okay Hopefully that shows up. Okay. Yeah, that's not bad Okay, so that's for any specific term we're looking for, right? So if I want the fifth term The fifth term is the k value of four one two three four five This is the fifth term k value Is one less the term you're looking for so the k value is four, right? So this is going to be c and it's to the power of seven seven choose four Okay, and it's going to be x to the power of seven minus four As for our formula and y to the power of k, which is going to be four so seven choose four is seven factorial over seven minus four Factorial times or divided by four factorial This is going to be x to the power of three y to the power of four, right? And the fifth term we already have the expansion here one two three Four five So this is the term we're trying to get out. We have the variables to the right power Now we just have to figure out if this guy Is 35, right? And the way we do that is seven factorial is seven times six times five dot dot dot over seven minus four factorial is three factorial And four factorial. Well, let me write this out properly Or the expanded version, right? Three times two times one and four factorial is four times three times two Now four three two one kills four three two one past here Three times two is six, right and six kills six so seven times five is 35 Right That is the same as that and instead of expanding it manually Right, which was a lot faster than Expanding and by foiling stuff out, right? We use the formula to find the fifth term which was faster finding specific terms pinpointing things, right? Knowing all of this now We're going to use the formula. We're going to erase Pascal's triangle And we're going to stick with the formula takes up less space And we're going to make this a little bit more complicated Binomial expansion and we're going to see how things play out for this, right? So let me erase all this Okay, now keep this in mind. I'm going to erase this part Okay Actually, maybe we'll keep Pascal's triangle out for now Okay Keep this in mind That's super powerful And I'm going to write this down here. So we see this, okay better See and choose k Okay x to the power of n minus k And y to the power of k Okay, let's erase this guy. So it's bigger. So we see everything and before we do a more complicated one I'm just going to read some more comments Just to make sure there's no additional questions related to this or any corrections In the way I've written things down Okay And choose k is most often written like an over k inside parentheses Yeah, the one comment is being made this Okay And choose k One of the ways that Is written Common certain parts of the world is Nk When they write it like this, that's the same thing as this Um, I usually write it like this I don't Usually write it like that, but that's one of the conventions that they use because that's referring to combinations I don't know if there's because you have another version of expansion where If you're doing permutations combinations is order doesn't matter here, but order matters here I don't know if there's another way of writing this For the p because you can't write n k for this as well because this means something else the expansion here Obviously, it's not that is n factorial over n minus k Factorial, but this one means n factorial over n minus k Factorial and k factorial So the difference between this and this is in the bottom of this one There's also a k factor in the bottom of this one. There is no k factorial And the difference between them is for this one order Doesn't matter no matter I'm sorry no matter This one order matters Okay So if order matters if you're dividing by one less term then you're going to have a bigger number Right, so there's more combination If order matters if you can tell the difference between the different objects, right if you're ordering 10 chairs, right and all the chairs look the same then order doesn't matter If all the chairs are different right different color different size Then where you place the chairs matters because you can distinguish between those objects, right? And that's something that comes up in Permutations and combinations which kicks off into probability statistics and stuff like this So let me erase this guy and this means this okay When we're expanding in the previous live stream that we did Earlier this morning we talked about this this means this okay, so we're going to keep that in mind as well So let's assume we had a binomial which was more complicated than x plus y. Let's say we had this Where should we write this? Let's write this here Let's say we had three x squared y minus five x y q to the power of eight Right wow wow wow way more complicated And I remember when we expanded this thing the x and y were basically your first term and your second term, right? So I don't know if they use x and y in the expansions. What do they use a's and b's or something? What did they use I don't even know Oh That's probability That's the thing that they're using so we can call it anything we want really But I'm I'm just going to leave them as x y but just know x means the first term and y means the second term And if we want to clear this up, maybe we'll do it this way We'll call it a and b all right b a means This and b means this Right and always remember the sign in front of the number Goes with the number, right? So this isn't five x y cube. This is negative five x y q, right? Just for example, if I had a negative number here, this is positive. Let's make it a negative, right? So what if for this? I wanted you to find the middle term right, this isn't Even power so there's going to be nine terms in the expansion, right? So let's find the middle number here. So what would the middle number be, right? If there's nine terms here, let me put these pens down for a second If there's nine terms in the expansion, right? So one two three four five six seven eight nine the middle term Is the fifth fifth term Right, and what does the fifth term mean the fifth term has a k value of four Right because the k value here is one less than the term, right? So let's assume We want to find the fifth term which gives that a k value of four again, right? So the way this expansion will work is this This is going to be eight the power Right n refers to the power eight Choose four Right, and then we're going to take this guy, which is negative three x squared y And it's going to go to the power of n minus k, which is eight minus four So it's going to be eight minus four right And then the second term the b term is going to be negative five x y q negative five x y q to the power of k which is four interesting cool Difficult possibly but let's do it again. Okay, this eight choose four is going to be eight factorial over eight minus four factorial times four factorial on the bottom This guy is going to be negative three x squared y to the power of eight minus four is four And this guy is just going to be negative five x y q to the power of fourth Okay Now if we're going to expand this eight factorial is I should leave myself way more room. I'm going to erase this again and put it up here. Okay, so Let's do Actually, let's do this part here. Okay eight factorial means eight times seven times six times five times four divided by Eight minus four factorial is four times three times two dot dot dot Four factorial is four times three times two dot dot dot So four three two one kills four three two one here eight Four times two is eight right So that kills that three goes into six twice Right, so in the top we have seven times five is 35 times two is 70 okay So can you see this far down? Oh, I didn't change the view my bad Jesus louis I think the camera is wrong. I think it's wrong. I should have checked the back check the screen So let me do this part again my bad So this one was going to be eight seven six five four dot dot dot over Eight minus four Right. Yeah, we've got the right view eight minus four is four Factorio this one four factorial. So four times three times two times one dot And four factor is four times three times two times one four times three times two one kills this four times two is eight takes out eight three takes six down to two Seven times five is 35 times two is 70, right? So this is 70 Okay, and if we look at this expansion we took this to seven and if we look at the expansion for eight There's going to be one here. This is going to be eight That's going to be 28 add those guys up is going to be six 56 and add those guys up is 70 that's what we're finding right there those two guys add it up, right? So we have this for part that's 70 Now what we got is We've got to expand this guy. So let me expand this guy. So we have the 70 Let's erase this So we know this guy that's 70. I'm going to write this down here 70 times this That's going to be negative three negative three x Square y to the power of four That means negative three is to the power of four x squared is to the power of four and y is to the power of four So negative three to the power of four is negative three times negative three times negative three times negative three Which is going to be three times three is nine three times three is nine nine times nine is 81 And it's even number of negatives so it's positive So that's going to be 81 x squared to the power of four is x to the power of eight y to the power of 1 to the power of 4 is y to the power of 4. So this guy is going to be, this guy becomes 81x to the power of 8 y to the power of 4. Now let's expand this guy. That's negative 5, negative 5x y cubed to the power of 4. Negative 5, so negative 5 to the power of 4 is 4 5 multiplies together. 5 times 5 is 25. You can do it. Multiply it by 5 is 125. 125 times 5 is 625. Or you could go 5 times 5 is 25. 5 times 5 is 25. 25 times 25 is 625. It should be anyway. So this guy becomes 625. That becomes x to the power of 4 and y to the power of 12. So this guy expands to 625 x to the power of 4 y to the power of 12. So what does this give us? I'm going to erase this. Now I'm not going to multiply the numbers together. You can do that on your own. But it's 70 times 81 times 625. Actually let's do it with the calculator. Let's do this. If anybody wants to, they can do. Let's go 70 times 81 times 625. 3,543. So 3, 5, 4, 3, 5, 4, 3, 7, 5, 0. And the x's and y's combine you add the exponents. So x to the power of 8 times x to the power of 4 is going to be x to the power of 12. y to the power of 4 times y to the power of 12 is going to be y to the power of 16. So the middle term for this expansion is 3,543,750 x to the power of 12 y to the power of 16. Just imagine having to expand this manually by hand eight times. That would be crazy, it would be huge. A lot of work, a lot of work, a lot of work. Using this formula, this formula right here, we can go to an exact term in an expansion, binomial expansion. Very powerful, very powerful. And what they end up doing is they give you these types of questions and they try to get you to find certain terms. They could ask you to find a constant term in an expansion. So let's do one where they ask you for a constant term. Let me erase these guys. We'll keep Pascal's trying a lot. Let's erase these. And I'm going to have to give you a binomial where there will be a constant term popping up. So it would be something like this. Let's assume it was 2x squared. Let's make sure we can get one with an expansion minus, with a constant term, minus 1 over x cubed. I think that should do it. Should. If it doesn't, we'll change it up. Let's try it out. Let's get you to the power of six. Okay. Let me make sure there's no more comments. I'm going to read a couple more comments. If there is any. Yeah, that's the camera. Okay, cool. No comments or no questions. So let's expand this. Now if we're doing this guy, one question they love asking when they're giving you guys Pascal's triangles, binomial theorem and expanding binomials and stuff like this. Sometimes if you're unlucky, you get a teacher saying they want you to prove that you understand how this binomial expansion works. They'll ask you to expand this whole thing, which isn't that difficult, right? Expanding this whole thing would be, again, the first term would be 6 choose 0, 2x squared to the power of 6 and negative, remember, 1 over x cubed to the power of 0 plus c choose 1. We kick it up. The k value goes up by 1, right? And then it would be 2x squared to the power of 5, which would be 6 minus 1 and negative 1 over x cubed to the power of 1 plus so on and so forth the way we laid it out. But if they want the constant term, this is what's going to happen. If you notice here, x squared is in the numerator and you got an x cube in the denominator, right? So when they ask you for the constant term, because you're multiplying things out, your first term a and your second term b, because we have x in the top and we got x in the bottom, the x's will cancel each other out. So when they ask for the constant term, what they want you to find, they want you to find the term where these x's kill each other, right? And there's no x's in the term, it's just a number, right? So for you to be able to do this, you're basically looking for common denominator between these. You want these to be the same power, right? So the common denominator between these is 6, right? If x was to the power of 6 up top and x was to the power of 6 in the bottom, then they would kill each other. All you would be left with would be just a number, right? Let me just read one more comment. It's being held for review. Allow, we'll post it to check. Yeah, we're gonna ban this guy. Okay, let me do a minor banning because we don't have any mods right now, right? So let me take those out. A little deviation about this guy. How do you ban? Okay, we do manually. Let's do another one. Oh, he's got two names. How silly. Cool. Apologies about the troll action, right? Sorry, but I don't understand that. Can you give me a short introduction to binomial stuff? We just did an hour intro to it, right? So the stuff is there. We sort of built this up, right? But what we're gonna do now is do the expansion for this, but just find the constant term. Okay, hopefully that's okay. The binomial expansion is pretty cool, actually. It's very powerful. Saves you a lot of time. Saves you a lot of time. And I'll check the chat more often just to see if there's any more hordes of trolls or single trolls coming through. I think he's lost. Okay, thanks. So let's find the constant term for this, right? So this sometimes becomes trial and error. You can guess approximately where it is, right? That's x squared. That's x cubed. Common denominator is 6. So if we can get this to a power of 6 and that to a power of 6, then they'll kill each other. The only thing we're left with is the numbers, right? So let's just at random try something. This is the 6th power. So this is the expansion we're gonna get for the c value, right? Those are the coefficients in the front. Okay, so let's just for random try to find the, I'm not gonna try to find the third term. Actually, let's try to find the third term. The third term would be the k value of 0, 1, 2, right? So let's try to find, yeah, let's do the third term first because it's not going to work. It's not going to be a constant term. So the third term is going to be c. Let me write this over this way. So we have rho. It's going to be the power of 6. If we want the third term, it's one less than term number, the k value, right? So we're going to put 2, and then we're going to go 2x squared to the power of 6 minus 2, which is 4, and then negative 1 over x cubed to the power of 2, right? This, we already know what it is. It's the third term. So we know that's going to come out to 15 here, right? We can do it. Just check it. It's 6 factorial over 6 minus 2 factorial, minus 2 factorial, and 2 factorial, right? 6 minus 2 factorial is 4 factorial, right? 4 factorial. This is 6 factorial, 6 times 5 times 4 times 3 dot, dot, dot. 4 factorial kills these guys. 2 factorial is 2. 2 goes into 6. 3 times 3 times 5 is 15. So we already know what this is, right? That's 15. This guy, 2 to the power of 4, 2, 4, 8, 16 is going to be 16. x squared to the power of 4 is x to the power of 8 times x cubed squared. It's a negative being squared, so it's positive. So in the bottom, we're going to have 1 over x to the power of 6, right? And if we simplify this, right? 15 times 16. What is that? 15 times 16. 15 times 16 is 240. So this guy is 240. x to the power of 8 times 1 over x to the power of 6, and x to the power of 6, these guys kill each other, right? So this reduces this down to 2, okay? So the third term is 240 x squared, okay? That's not a constant term because we still have x's in there. So we had one more, we had x squared in the top, right? So we have to move along this thing because as we move along the binomial expansion in this direction, the power on the A comes down, the power on the B goes up. So what I'm going to do, I'm going to find not the third term, but the fourth term, okay? And hopefully that will be the constant term. If it's not, this thing doesn't have a constant term, okay? So what we're going to do now is do this, but we're going to go 6, choose not this. We're looking for this. That's the fourth term. So we're going to choose 3, okay? And we should have sort of known that, right? Because this is n minus k, 6 minus 3 is going to be 3. Actually, it's not going to work out. We're not going to have a constant term and pick something that doesn't work. I think so anyway. But this is going to be 2x squared to the power of 6 minus 3 is 3. And this one is going to be negative 1 over x cubed to the power of 3, right? The number, this number is 20, that guy. 2 cubed is 8. 8 times 20 is 160. So we took care of the coefficients. That's going to be x to the power of 6. That's going to be 1 over x to the power of 9. So this guy simplifies down to 160 times x cubed in the bottom. So we've already gone past the point of I did this correctly anyway. I hope so. If this was a 3 and that was a 2, it would work out, I believe. So this expansion doesn't have a constant term because there's always going to be x's in there. Let's change these guys up. Let's make this a 3 and make that a 2. Just so for our satisfaction, 4, then it's not going to be that term. It's going to be 1 more. So if we do it that way, if that was, where were we? Oh, I was going to change this to 3 and change that to, that'll be 3, that'll be 2. So that's to the power of 9 and that's to the power of 6. That doesn't work. If it's to the power of 2, no, it's still not going to work. I think something that doesn't work. That's weird. Yeah, it doesn't work. Sorry. Okay. But basically what happens is you can do this, get expressions like this where the terms end up killing each other. The simplest way to do it is just keep these as 1. If we keep these as 1, then finding, oh, because I did this wrong, this should have been 6 minus 2. That's the reason. Silly me. 6 minus 2 is 4 and that would have been 2 and that was a 2 and a 3, 2 and a 3. That was an 8. That was a 6. That didn't work. Oh, no, we did that already. Okay, that didn't work out. But if the simplest way to do this is, if that was x to the power of 1 and if we end up finding the fourth term, this becomes 3 and this becomes 3 and that becomes 3 because the 6 minus 3 is 3. So this would be the third term, the fourth term, this would be 20. Here, let's do this expansion for this guy. Let's make sure we don't have any more comment. Sorry, but I don't understand this. Well, it would work if you just switched. If you multiply everything by x to the 18, you can move it inside and get... Yeah, you could actually. But let's try this one out. Let's make it more simpler, right? So this would be 6 choose 3, right? So it'd be 6 factorial over 6 minus 3 factorial, 3 factorial. So this is 3 factorial. 6 factorial, 6 times 5 times 4 times 3 dot dot dot. That kills the factorial for the 3. 3 factorial is 6. So 6 kills 6. 5 times 4 is 20. So that's that guy, right? So that guy is 20. This guy becomes 8, right? That guy becomes x to the power of 3. This guy becomes 1 over x to the power of 3. x to the power of 3 kills x to the power of 3. And 8 times 20 is 160. That's your constant term. So the constant term occurs is the fourth term in the expansion of this, which is to the power of 6, right? So the fourth term is your constant term, which is 160. Sorry about mucking that up the first time. Get go, right? The banning things throws me off a little bit when I'm banning people. I don't like banning people. It's weird to me, right? People don't behave. The troll's coming through. I guess they're having fun. Okay. Let me put out my glasses. Let's check this out. That, that, that, that. Well, it worked if you just switch. If you multiply everything by x to the power of 18, you can move it inside and get 2x to the power of 6 minus 1 x that, or it would be 2x to the power of 5 minus 1 to the power of 6, if I remember right. And it's easier to see that there is no x to the power of 18 term. x is not a multiple of 5. Okay. Yeah. I, I should have prepared something. I, I was thinking there might be questions that wouldn't go down this route. I did one of these things, a couple of these things. I had some students that are taking commentaries, so we did some reviews with them. And they have a whole bunch of, these types of questions, whole bunch of different ways they come at you. Because there's only so many times, so many ways they can ask you the same question, right? So they try to mix things up and mixing things up, mixes up. Chico doing the mathematics, so I make little mistakes here and there. What I mean is the constant term of x squared 6 is the same as x to the power of 18 term. Is it 2x? But there has to be an x in the denominator. Let's see that. I think next time I'll try to come up with a better example. I'll lay some stuff out. I should have actually a little booklet that I have examples written down, but I don't. I don't. I just, I like, because I teach students from different schools and stuff like this. So each teacher teaches them differently. And there's a lot of bad teachers out there, unfortunately. It's just the system the way it is, right? That one, 2x to the power of 5 minus 1 to the power of 6. Yeah, x to the power of 18. Let's check this out. x to the power of 18 times 2x here, that equals, oh, does it? Okay, come on, you did the expression. You did the thing on paper. That's cool. I like the speed that you do math in eraser. You're very good at it. You have a nice grasp of the language of mathematics, the syntax of the language, which is fantastic. Which is fantastic. 1x to the power of 3 per factor. 1x to the power of 3 per factor. I'll have to think about that. I'm losing it on that one. I have to, for me, one thing with mathematics is I have to see it. I'm very visual with it, as you can tell, obviously, right? So I have to sit there and expand them and all of a sudden I go, ah, it just makes sense to me, right? I think that's one reason my students like working with me because I don't assume anything. I always, sometimes the most simple things I have to work out, just to say, oh, wow, okay, that works out that way. That works out that way. Yeah, that's correct. x to the power of 3 times 2x squared minus 1 over, yep, that is correct. That works. Reading math in the computer. Yeah, for me to reading math in the computer is so difficult, so difficult. As for sweetie's question, how are permutations and combinations applied in the real world? In probability and statistics in a big way, right? Because if you're trying to find the probability of something, you have to find out what all the possible outcomes of the situation are before you can figure out what the probability of a certain type of outcome for a situation is, right? Permutations and combinations comes in, comes in playing a huge win when it comes to gaming and poker, right? Like, for example, what's the probability of getting two pairs in poker, right? So let me erase this. And that's one thing in, for mathematics. Oh yeah, makes sense. That's one thing in mathematics. It's probability and statistics came out of games of chance. So let me change the angle here. Just to give you one quick example. For example, what's the probability of getting a two pair in poker, right? So you have five cards in poker. So one, two, three, four, five, right? And you want to pair these up, right? Let's say you want a three and a three and a 10 and a 10 and you can get whatever you want here, right? It doesn't make a difference, right? Now there's 52 cards in a deck of cards and there's 13 cards per suit and there's four suits, right? So if I'm doing this correctly, I think racer, racer killer can correct me on this. But the probability of getting two pair is this, or the number of, not the probability, but the probability would just be the number of combinations divided by the total number of outcomes in a poker game. But the number of ways that you can make a pair in poker would be 13 choose two. You're going to take 13 cards, right? And choose two and then from there you're going to, because there's four cards per, what do you call it, number, I guess, like there's four threes, there's four tens, there's four jacks, four choose two. So for each one of the 13 cards from the suits that you chose, you're going to choose two. I think that's what it is. 13 choose one, four choose three. Or wait a second, is that going to be... Oh man, we just did this recently, but we're doing so many poker stuff. I totally forgot what it ends up being. Or maybe is it this? It's 13 choose one. So from the 13 from...