study material-physics

# Simple Proof of Binomial Theorem for IIT-JEE

__THE PROOF OF BINOMIAL THEOREM__

__ __

Well most of the books like Arihant and all have presented the proof of Binomial theorem but either it is too difficult or the theorem is stated as such. I have given a very simple rule which makes the theorem look like a COMMON SENSE plus I had described why we have things like ^{n}C_{r} appearing as coefficient.

Consider the binomial expansion of (a + b)n

So indeed we have the cases

(a + b)(a +b)………n times

So on multiplication we are free to form combination of a and b so that they have total n terms. I mean to say the sum of the coefficient of a and b is n . So the general term will be of the form

A a^{r} b ^{n-r} ……….1)

so like this many combination are possible . For example consider (a+b)2

so (a +b)(a+b)

Let revise class 4 th concept. How do we mutliply we take one part of first bracket and mutliply it to second bracket and so on…….so we get

(a + b)(a + b) = aa + ab +ba + bb

Moreover we can say that

“ __NO TWO TERMS AFTER ADDITION WILL CONTAIN SAME COEFFICIENT OF a__ “ ……..2)

Like we can never have terms like

(a + b)^{n} = ………A a^{r} b ^{n-r} + B a^{r}b^{n-r-1}

^{}

So now a simple combination is need ……

Consider a visual picture

(a + b)^{n} = (a +b) (a+ b)…… n times

Now friends consider each (a +b ) as a bag containing two balls marked ‘a’ and ‘b’. so we get

(a + b)^{n } = { a , b } { a,b } { a,b} …… n times

Now all you have to do is select n balls so that “ NO 2 BALLS ARE FROM THE SAME BAG”

And from the result 2) if we take r balls marked as ‘a ‘ so n-r are blue so focus on one ball say ‘a’ .

So tell me from n balls we need to take r balls so what is the total number of ways of seleciting these balls. Yes it is the nCr so the

Term a ^{r} b ^{n-r} appears nCr times so the coefficient A is nCr.

Moreover we have terms like

a^{n} and b^{n} so the value of r is from 0 to n.

Hence we get

(a + b)n = Σ_{r=0 }^{n n}C_{r} a^{r} b ^{n-r}

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