 Hello, everyone. How are you? So here I am back again with yet another question on Binomial theorem. You can also call it to be a multinomial theorem question because it has got mixed up both the words But before I say anything else, I would like you all to look at the question first Now I would request you all to pause the video right here and Try to solve this question on your own Alright, I hope you have made an honest attempt to solve this question Now let's discuss the solution So for solving this question We would need our basic understanding of binomial theorem and of course a bit of multinomial theorem as well Don't worry about it. I've provided you the link for these of my video lectures in the description box right below So we are going to start here by assuming that this expansion is actually a Binomial expansion with a fractional power Okay, so I'm going to write down this expression as 1 plus the rest of the terms Right. All right, and think as if this is like, you know 1 plus Some capital X raised to the power of 2 by 3, right? So we all know how to explain a binomial theorem with the fractional power So that goes something like this 1 plus 2 by 3 capital X Plus 2 by 3 times 2 by 3 minus 1 upon 2 factorial x square and so on Right now since this is having a power which is a fraction this Binomial expansion is never going to stop. It's going to go on and on forever, right? So in plain and simple words if I have to ask what is going to be The p plus one-th term of this expansion, right? I'm very sure that your answer is going to be 2 by 3 times 2 by 3 minus 1 2 by 3 minus 2 and so on till 2 by 3 minus p plus 1 upon p factorial x to the power of p Right So the general term or you can say the p plus one-th term of this expansion is going to look like this alright So let's put our x back that is going to be 2 by 3 2 by 3 minus 1 2 by 3 minus 2 and so on till 2 by 3 minus p plus 1 and The x that we are talking about is this, you know lend the expression that you see over here That's negative of 3x minus 2x square and plus 6x cube Whole to the power of p, okay, and this entire thing is going to get divided by p factorial Right Now let me tell my viewers that if I want the coefficient of x to the power of 3 in this expansion What I'm going to do is I'm going to assume that let x to the power 3 term be obtained Obtained from TP plus one-th term Okay, so what I'm claiming here is that this term that I have written in white over here Right, this term is responsible for giving me x to the power of 3 okay now Let us now focus more on This term which I'm showing with this zigzag in the base Okay, so let's talk about that term separately. So minus 3x minus 2x square plus 6x cube To the power of p right if I write the general term for this Okay, so let us write down the general term for this. So for this expression the general term is Given by P factorial upon Alpha factorial beta factorial gamma factorial Minus 3x to the power alpha Minus 2x square to the power beta and 6x cube to the power of gamma right Now let us say this general term is the one which is actually giving me x cube Right, so what can I say over here? I Can say here that number one The powers of x which happens to be alpha To beta and 3 gamma they add up to give you 3 that's number one thing I can say and Another thing which I can say here is that Your alpha plus beta plus gamma will actually add up to give you a P Now you can refer to my multinomial theorem discussion Whose link I have provided in the description to know more about it Now what I'm going to do is I'm going to solve for Alpha beta gamma from this expression by taking a guess at these values because it's easy to guess them as Alpha beta and gamma. They are all whole numbers. They belong to whole numbers Okay So if I do that, let's have I know the possibilities Written over here, so let me write down them in the column mentioned here alpha beta gamma Right, so let's start by putting gamma as zero So when I put gamma as zero and let's say beta also as zero alpha becomes three, isn't it? So basically I'm trying to satisfy this equation, which I'm showing with the double tick mark And if I take my gamma as zero, I could have beta as a one and gamma also as a one Okay, and finally if I take my gamma as a one, then my alpha beta are bound to be zero zero each So I feel these are the only three cases in which you will be able to get Alpha plus two beta plus three gamma equal to three All right Now for these expressions, let us figure out the value of P So for this the P value is going to be three plus zero plus zero. That's a three For the second case, it is going to be one plus one plus zero, which is a two and for the third case It's going to be zero plus zero plus one, which is a one. All right now This term is actually a part of the bigger term TP plus one, right? So let's go back. Let's go back to the expression Proficient of x to the power three is going to be two by three Two by three minus one Dot dot till two by three minus P plus one upon P factorial times P factorial alpha factorial beta factorial gamma factorial Minus three x to the power alpha Minus two x square to the power beta six x cube to the power of gamma Okay, so I just realized I wrote x in those expressions By the way, just ignore the x's which I have written that is minus three x Just read it as minus three minus two x squared Just read it as minus two and six x cube just read it as six because in the coefficient There will not be any variables, right? So I hope you can understand that Now we have only figured out these three cases where I can actually get the value of x to the power three Right. So these are the only three cases of alpha beta gamma which generates x to the power three So let's write down those three cases. So the first case is where my P is three Alpha is also three Beta and gamma are zero each. So this is your first case. So one of this your coefficient comes out to be two by three Two by three minus one and mind you when you put P as a three it goes to two by three minus two which is actually the very next term Okay, and divided by P factorial Which is going to be three factorial into three factorial which coincidentally would get cancelled off three factorial Zero factorial zero factorial and here you will have the coefficient as minus three to the power of three and The rest of the terms are going to vanish Right because beta and gamma are zero. Okay, so let's evaluate it finally. How much does it give you? So it comes out to be two by three times minus one by three times minus four by three times minus 27 upon six So this simplifies further to minus four by three All right, so this was my case number one Let's talk about case number two case number two is a scenario where my P is two Alpha beta are one each and gamma is zero. So I'm talking about this scenario these two alpha is one beta is one and gamma is zero So let's take that up So these two Alpha is one Beta is one and gamma is zero. All right, so let's see. So what's the coefficient coefficient is going to be two by three and The next term will be two by three minus one and I think this is also the last term divided by two factorial Okay times two factorial one factorial one factorial zero factorial and I'll have minus three to the power of one and minus two to the power of one Okay, so I think this further simplifies to negative four by three Alright, so with this we move on to the third case So third case is What we see here at the bottom where P is one alpha beta zero zero and gamma is one So let's look into that case So P is one Alpha is zero Beta is zero and gamma is a one So in this the coefficient of x to the power three comes out to be two by three by one factorial into one factorial by zero factorial zero factorial one factorial Six to the power of one. So this was down to four All right, so let's now club all the coefficients together So the combined coefficient of x to the power three here will come out to be minus four by three minus four by three plus four and That's nothing but four minus eight by three. That's actually a four by three Okay So the answer to the question is the coefficient of x to the power three is four upon three So guys here you are So I hope this video helps you to revise your binomial concepts This question seemed to be easy, but it wasn't it involved a lot of concepts for us like the general term the multinomial expansion And I hope you really learned something new from this. Thank you so much for watching. Stay safe. Stay healthy