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Simple proof of Binomial Theorem
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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 givena very simple rule which makes the theorem look like a COMMON SENSE plus I had described why we have things like nCr appearing as coefficient. Follow me : 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 ar 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 ar b n-r + B arbn-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 an and bn so the value of r is from 0 to n. Hence we get (a + b)n = Σr=0 n nCr ar b n-r So we have done what Sir Isaac Newton did in his own time……. Blade- X // May 30th , 2009 |
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