V : jee main /advance class 11 mathematics : Binomial Theorem class-1 ( 29 may 2026 )

Añadido: 2026-05-30

[00:00:32]Good evening sir.

[00:00:34]>> Good evening.

[00:00:37]Let's start a new topic.

[00:00:42]>> Okay, sir. Let's start binomial theorem because uh actually it's so much related with uh it's very much related with sequence and series and sometimes most of the times when they ask questions now they actually mix up the questions of binomial theorem with sequence and series.

[00:01:05]So we will try to do sequence and series because we have sorry we have completed sequence and series and now we'll try to do binomial theorem because actually this is also a form of series only binomial theorem.

[00:01:23]Okay.

[00:01:27]>> Okay sir.

[00:01:34]The binomial theorem.

[00:01:37]One minute. Just one minute.

[00:02:11]Yeah.

[00:02:14]See what is there in binomial theorem.

[00:02:37]Okay.

[00:02:39]Binomial theorem basically is an expansion of x + a power n x + a power n when it is expanded it's written in the form n c into x power n a power 0 + n c1 x n -1 a power 1 + n c2 x n - 2 a power 2 so on and so on and so on. n c n -1 x power 1 a power n - 1 + n c n x power 0 a power n clear it goes like this >> okay sir >> now this is the basic expansion and in this later when you will come to know about per come to learn about the permutation combination You will know what is NCR actually this the term that you are able to see here now this NC1 NC2 NC minus one okay let me first define this NCR this is actually if we talk about the meaning it's actually combination it is number of what is it represents number of ways of select ing our things out of n different things.

[00:04:33]Okay, it's but if you do not understand this one then don't worry also because in when we'll come to the permutation combination we will discuss this in detail. It's the meaning. Now what is important to us to know is the formula of NCR. NCR is actually written as N factorial by R factorial into N - R factorial. Right?

[00:05:03]What is the meaning of factorial? Let me define few things here. One thing is there n should also always be greater than zero. R should always be greater than or equal to zero.

[00:05:19]That's there means the combination the NCR this formula is actually defined only when n is greater than z and r is greater than z. for negative number that is not defined only.

[00:05:32]>> Thank you sir. Now one peculiar thing to understand peculiar thing to observe here uh later I'll tell you about what is this fact this n this is known as n factorial n factorial r factorial r factorial let me tell you now only n factorial means You are taking the numbers 1 starting from 1 1 2 3 4 5 all you are multiplying and taking up to n.

[00:06:14]If I talk about you understood now if I talk about 3 factorial it will start from 1 multiplied with 2 multiplied with three meaning is 6. 2 factorial means 1 into 2 that is 2 only. 4 factorial means 1 into 2 into 3 into 4 >> 2 4 okay >> 24 so what's the meaning of factorial factorial means the number that you are taking with the factorial that number the numbers will start from one and they will move up to n and all numbers will be multiplied that's the meaning of now the one better the what are the observation here when You see here this expansion this is actually very important using this only you will be solving all kinds of questions.

[00:07:10]If we talk about the if we talk about the expansion of x plus a whole power n which is known as binomial theorem series expansion if you see properly the terms are starting from zero mean the zero is the first term able to see now.

[00:07:30]>> Yes sir.

[00:07:31]>> When here this is zero this is our first term. When this is one, this is our second term. And when this is two, we are getting this third term. So actually when we'll move up to n, it will be what? n + 1 term.

[00:07:48]>> Okay sir.

[00:07:49]>> So total number of terms total terms in this series are actually n + one that's one thing.

[00:07:59]So that means if I have given you you can check this on check this in our normal cases also if you see >> similar to AP sir >> uh similar to AP >> uh but the problem is this AP what's the difference between this and this in AP we have a common difference we have a common common difference between the second term and this first term but here there's no common difference here.

[00:08:29]>> Okay.

[00:08:42]Okay. Now coming to the proper observations this NCR if you see this is NCR. Now if I talk about get me the value of n c n minus r you have to put do it by yourself you have to put n minus r in place of r and then give me the formula and then see what you are Sir you said I have to put n minus r in the place of r.

[00:09:32]>> Uh n minus r then only you'll get the formula of n cn minus r.

[00:09:39]Yes, >> that yes >> just in the in the formula that is given to you NCR put the value of n put the in place of r you have to just put n minus r and then see what formula you're Okay.

[00:10:01]Sorry, >> sir. N² + R² - 2 R.

[00:10:15]>> How?

[00:10:18]See in the formula you will be putting n factorial above numerator. Correct?

[00:10:23]>> Yes sir.

[00:10:24]>> In place of r you putting n minus r factorial.

[00:10:28]>> Yes sir. Into n >> into uh into n minus r means you'll be having n - n - r.

[00:10:38]this >> this will become actually n factorial by n - r factorial into n - n + r whole factorial and it will get cancelled. So actually you're getting the formula is n factorial by n - r factorial into r factorial right?

[00:11:00]>> Yes sir. What the thing of observation is the formula for NCR was is was also same. So the formula of NCR the formula of NCN minus R are actually become becoming same correct.

[00:11:19]>> Yeah the denominator values are reversed actually.

[00:11:22]>> H but the value will be same. Now suppose if I have in place of R you have something like uh three or some number.

[00:11:31]So if you take that same number you have this number particular number and then you calculate the value that r factorial suppose that is 3 factorial 3 factorial will be 1 into 2 into 3 and there also it will be 1 into 2 into 3 the values will be same now final answer would be same >> yes sir >> so what we are actually observing first observation we are having is first observation was this total terms is equal to n - 10% Second observation we are getting is ncr is becoming equal to n c n minus r.

[00:12:09]What's the meaning of this? The meaning of this is the in the expansion when we put r when we put r is equal to zero. Hm nc not is the first first one coefficient of first term this will be equal to n cn minus0 means n cn meaning >> okay >> this the first coefficient of first term and the coefficient of last term are equal and then consecutively when when you move back from front side you will move and then from the back side you'll move you'll actually get the same value means the value of nc CN minus 1 will be equal to the value of NC nc1. Similarly, if you go NC cn minus 3, NC minus2 value would be equal to the NC2 cn2.

[00:13:03]>> So what's happening now? The only this x powers and a a powers are changing the coefficient value is actually same.

[00:13:13]Understood this second observation.

[00:13:18]Now this is your basic series which you should actually remember. If you want you can write on your copy that's important. Now if I ask you uh let's can you give me the formula for x - a power n can you suggest me what you will do?

[00:13:43]You have the same >> instead of writing plus >> ah you had n c into x power where from where we started x and we had >> n - a power 0 plus n c1 x power n -1 - a power 1. So it will go like this then n c n -1 x^ 1 into - a power n - 1 and then >> n c n x^0 - a power n like that >> simple >> now understood now if I give you >> yes uh remember that remember that series the original series.

[00:14:41]Now if I ask you can you give me the value of what is the sum of series n c + n c1 + n c2 + n c3 so on and so on n c n -1 + n cn can you give me the value of think then you will then after this I want you to find out the NC KN + NC2 + NC4 4 + NC C6 so on and so on this is sum of all even coefficients >> but I'm not able to find out >> not able it's very easy You have I'll give you an idea.

[00:16:56]>> Okay sir. After getting this you'll be able to do all you have the expansion of x + a power n as n c into x power n into a power 0 + n c1 into x n -1 a power 1 + n c2 x n - 2 a² so on and so on up to n cn n x^0 k a power n correct?

[00:17:33]>> Yes sir. I want the sum of this only means what I'm concerned I'm concerned with these only right I I'm not concerned with other other things x and a >> so what I will do >> and c not >> so what I will do I will put x = 1 and a = 1 >> okay once you do x = 1 a is equal to 1 power of 1 is 1 only whatever power may be there. So you will be getting this sum n c + n c1 + n c2 + n3 n minus 1 and what it will be equal to x is 1 a is 1 1 + 1 power n that's what 2 power right >> yes sir >> now in the same way see now what I'm going to tell you whenever I'm asking you the sum of any series involving binomial coefficients you have to put some value of x and some value of a whatever you it's up to you whatever you want you can take to get me the desired answer desired series like we have done here Okay.

[00:19:25]Give me the answer.

[00:20:38]Sir is 2 ^ nus one 2 the^ nus why >> sir I actually thought of subtracting and do it >> then in that case how did you remove the all terms M >> you told 2^ n minus one now.

[00:21:14]>> Yes sir.

[00:21:15]>> Your answer is correct. It's not wrong.

[00:21:21]But I want to know how did you get to the answer.

[00:21:28]Sir, I subtracted the first equ like n c minus n c1 and done the same thing with uh n c2 minus n c3. Then I'll get zero.

[00:21:46]I got zero. H >> uh then I added n + n c2 + n c4 which equals 2 n + 0 then I got 2n by 2.

[00:22:08]>> See this is this total this thing you have already known. Okay. Now if I uh if I get this this you already know keep it as one.

[00:22:25]Now if I do x - a power n it is n c into x^ n - a power 1 0 + n c1 x power n - 1 into - a power 1 >> plus and so on n c n x0 into - a power n Yes sir.

[00:22:55]Now if you want see if I put x = 1 here and a = 1 here in that case what will happen this will become zero left part will become zero and this sum is equal to what this sum is equal to n c minus n c1 plus n c2 2 - NC3 plus and so on. N CN, right?

[00:23:30]>> Yes sir.

[00:23:37]>> Put it as second.

[00:23:40]Now I want to remove I want the sum of only even terms. So I will add 1 and two. 1 + 2 what I'll get?

[00:23:50]All odd terms will be removed and even terms will be added two times. Right?

[00:23:55]>> Yes sir.

[00:23:56]>> So it will be 2 into n c plus n c2 plus nc4 >> up to the last even term that is equal to 2^ n >> plus 0.

[00:24:08]Now I want the even terms only. So I'll divide two that I've got the left side.

[00:24:15]It will be two power and divide by 2 which will be equal to 2^ n minus one.

[00:24:19]Okay. This is your result.

[00:24:24]>> Okay.

[00:24:35]>> Now in the same way can you give me the result? This became your second result.

[00:24:45]This was your first result.

[00:24:48]Let me put it in the box.

[00:24:58]Now I want you to get me the sum of odd terms only. n c1 + n c3 plus n c5 plus and so on.

[00:25:13]What is this?

[00:27:35]s I'm getting the same as the above term 2 and 2 to the 2 power n minus one >> same only know because here the only difference is see you Add n c kn plus n c1 + n c2 the whole sum is equal to 2 power n right now the even terms why no I don't think so I don't think so you'll be getting same Sir I took x= a and x= to 1 and a = 1 >> m >> and then I subtracted aa okay okay you subtracted 1 - 2 now okay 1 1 - 2h correct only let me check 2^ n - Z uh correct correct correct. Let me check the answer.

[00:28:55]Uh both ways you'll get the same answer.

[00:28:58]Good. NC1 + NC2 NC3 plus NC5.

[00:29:04]Right? Very good. So this nc1 + n c3 plus nc5 the all sum is equal to 2^ n minus one.

[00:29:16]Very good.

[00:29:30]So for now these results are okay. these results are uh in the future I'll tell you to get more results. Uh I said do you know about uh the differentiation integration during bridge course uh last month they had taught us in physics few points differentiation I think >> aa later we'll talk later we'll study don't worry about it later I will introduce differentiation integration for this but not now.

[00:30:13]Now my next thing is what I want to tell you is term independent of of x.

[00:30:26]How to find the term independent of x?

[00:30:30]If you remember this series x^ a l n we have written n c into x^ n a power 0 + n c1 x^ n -1 a1 so on and so on up to n cn x^0 a power like that correct now if I and you also know this is your first term this is your second term Correct. This is your n + 1 term.

[00:31:04]Correct. Now if I ask you what would be the t r + 1 term? What would you say?

[00:31:20]What would be the r + 1 term? t r + 1 means r + 1 term.

[00:31:28]what it would be? Can you give me a general expression for the r term r + one term?

[00:32:04]s I'm not able to guess if this is first term for first uh let me write a general term ncr x^ n minus R A power R will it work? See because if I put the value of R is equal to Z you are getting first term.

[00:32:38]>> Yes sir.

[00:32:40]>> What's the value? n C 0 X^ N -0 A power 0. You're getting the first term right?

[00:32:48]>> Yes sir. Now if you uh want me to get the first term the for second term put r= 1 r= 1 means you'll be getting uh t r + 1 means second term >> what it will be n c 1 x^ nus 1 k 1 >> right?

[00:33:11]>> Yes sir. Okay.

[00:33:13]>> So this is the general term. Now what where you have to be careful is when you have the r + 1 term suppose I asked you to get me the third term or get me the seventh term h then what would be the value of r you will take >> n uh r is equals to 7 No, >> no. Six.

[00:33:46]>> Six. Correct. That only I want you to focus.

[00:33:50]If you have seven, if they want seventh term, you should put the value of R is 6.

[00:33:58]6 + 1.

[00:34:03]>> Okay.

[00:34:03]>> Correct. Then you'll get the seventh term. It would be n c 6 x^ n - 6 k a power 6.

[00:34:34]Sir, why do we use this binomial theorem?

[00:34:45]Uh binomial theorem is actually used to find out the different values of series.

[00:34:53]You know the for the uh the x + a whole square a + b whole square formula. How did you get it? You know how using a plus. [clears throat] [laughter] Uh >> correct. One way is you just multiply.

[00:35:13]Correct. But if we talk if we go on to increase the powers like a + b whole q a + b whole power five then it will become very difficult to multiply them.

[00:35:26]>> Yes sir.

[00:35:26]>> One way is multiplying them normally but the other way is through this. So to get the values of the to get the series which have the higher powers in which the sum of the two terms sum of two terms is to the higher power a + b whole power 5 or whole power 7 whole power 10 12 like that they have devised this binomial theorem and actually it is used in physics also in higher physics uh you'll see it's used basically to also also there There is one more use to simplify the series.

[00:36:07]Aa tell me you understood this one now the seventh seventh term. How to find out the term?

[00:36:13]>> Yes sir.

[00:36:14]>> Now what what was the heading? The heading was to find out the term which is independent of x.

[00:36:21]Now you have got a term that is t r + 1 is equal to ncr x^ n minus r into a power r. You know a is some constant. H.

[00:36:32][clears throat] >> Yes sir.

[00:36:36]>> Now if I ask you you have to find out a term which is independent of x. What you will do? Can you suggest me?

[00:36:49]>> Sir, it means that uh uh the other term should not be dependent of x.

[00:36:56]>> The term should not be dependent of of x. Yes. How you will how you will make this term independent of x >> sir? By dividing >> one is one way is dividing. Other ways if you if you somehow make n minus r is equal to z x^0 would be one >> one. Yes sir.

[00:37:19]>> So x will be gone.

[00:37:22]So to make what's the trick to get a term independent of x you have to put the power of x equal to z correct >> you have to put the power of x is equal to z. So in the in this particular case you have to put n minus r is equal to 0.

[00:37:45]So r value would be n. So n + 1 term n + 1 term is independent of x got it but it's not the same I'll give you a question in all kinds of question the results will not be same in all kinds of questions what you have to do in the particular question you have to just find out the power of x because always a is not constant A also may have a term which contain X.

[00:38:21]A may also contain X. In that case, you have to combine A with X and then find out the total power and then you have to make that particular power is equal to zero and then find out the value of R.

[00:38:33]Then R + 1 term would be your >> plus your V is also buffering your voice. I think there's some issue with the internet. Check.

[00:38:49]Uh I connected to 5G only.

[00:38:54]Uh 5G only.

[00:38:58]I don't want the back to right me offering Okay, let me give you one question.

[00:41:53]One minute.

[00:45:22]I'm not able to solve it.

[00:45:26]>> You're not able to solve this one. Check it.

[00:45:51]If 3^ 4^ 4k is a term independent of x in the binomial expansion, then k is equal to what? Hey.

[00:46:58]Perfect.

[00:47:44]What was there now? If 3^ 6 by 4^ 4 into K is a term independent of X in the binomial expansion of this, then K is equal to what? Uh let us try to find out R + 1 term. Okay.

[00:48:01]Now what is R + 1 term? would be 12 C R into X by 4 power uh 12 - R - 12 by X² power R correct?

[00:48:19]>> Yes sir. Now 12 C R try to find out the powers of X 1x 4 uh 12 - R it will be there X power 12 - R will be there uh divided by X^ 2 R into - 12 power R correct so it became 12 C R -2 power R / 4 power 12 - R into X power 12 - 3 R cor >> yes sir >> now I want independent term term independent of X so I have to put 12 - 3 R is equal to Z so R will be >> 3 R R= 4 >> R= 4 that means fifth term again I know in the fifth term there won't be any x.

[00:49:23]So they what they have given they have given the value of the fifth term independent term independent of x is 3^ 6 by 4^ 4 into k this value is actually equal to 12 c 4 into -12 power 4 by 4 power 12 - 4 8 >> okay >> which I can write 12 C4 into it's it will become plus only because power is uh uh even number so I'll do 12x 4 power 4 into 1 by 4^ 4 correct Yes sir. Okay. This will become 12 C4 3 power 4 by uh uh 4^ 4 C. But uh still the power six they have that means 3^ 6 by 4^ 4 into K is there 4^ 4 will cancel with this 3^ 4 will cancel here 3² so 9 K is equal to 12 C4 we have h I thought I will get the value in the let's see so k is equal to 1 by 9 into 12 factorial by 12 - 4 factorial into 4 factorial correct 1 by 9 into 12 factorial by 8 factorial into 4 factorial >> now can I write can I cancel with this and I'll get 9 10 11 12 because 12 factorial is 1 into 2 into 3 into 4 up to 12 >> 8 factorial is 1 to 3 up to 8. So 8 of that and 8 of above will cancel out.

[00:51:52]>> Okay.

[00:51:53]>> And here four factorial I can write as 1 to 2 into 3 and 4. Now 9 will cancel 10 into 11 into 12 >> by 4 >> uh divided by see because it's your first time that's why I'm going slow otherwise >> okay >> I would have done it directly 4 into 3 is 12 5 are 10 55 >> okay >> 55 should be your answer now did you understand how to solve these kinds of question.

[00:52:28]>> Yes sir. Can you go at the starting of the solution?

[00:52:38]Thank we separated xx 4 - 12x² whole 12. We separated that part and then we solved >> because see this became our first term that I use usually write with x. No.

[00:53:10]>> Okay.

[00:53:12]>> See actually I started with uh means I started with x plus a but I should have written like that a + b. It would become more better. You would better understand it.

[00:53:24]But here in place of x which we have taken x there it is xx 4 and here in place of uh a we have -2x².

[00:53:36]So just I replaced x xy 4 x with xx4 and a with - 12 by x² in the expansion in r + 1 the term and then I found out I found out the x power and then made it equal to zero so that I would get the independent term. Now value of independent term that is given to me is this 3^ 6 by 4^ 4 into k. So I equated the value of the independent term with that. Check here. This this is actually the core part of the solution.

[00:55:05]understood.

[00:55:08]>> Yes sir.

[00:55:10]>> Just try to understand it. Take your time.

[00:55:13]I'm going to give you one question of similar kind.

[00:55:23]try to do it at home and then uh I'll send you the uh worksheet also for this. Okay. Uh one more is missing I think a sequence and series did I give or not?

[00:55:39]>> No sir.

[00:55:40]>> Oh why you did not remind me.

[00:55:45]I'll send both. Okay. First you you will be able to do sequence and series full because uh the chapter is over after the class only exam.

[00:56:02]Take this as homework.

[00:56:04]This question should be your homework.

[00:56:15]Sir when are you when are they going to announce the results of 10th exam that you gave >> sir even I have no idea I have to check >> okay okay because I am curious actually I want to know your marks have proved because this time what happened in class 12th class 12th results so many problems are there actually it's auto checking they have checked and so many students are complaining it's wrong uh checking is wrong or correct?

[00:56:49]>> Yes sir. Even I saw it in the news also.

[00:56:54]>> So many problems.

[00:56:57]Let's end the present.