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Thread: Binomial Theorem Help

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    Binomial Theorem Help

    Screen Shot 2017-11-12 at 10.06.32 am.png

    Can you please explain the theory of the steps involved in solving this question, as I'm struggling with these types of questions. Also, any tips on proof questions relating to binomial theorem would be much appreciated.

    Thanks for your help.
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    Re: Binomial Theorem Help

    So it says both sides but to get the second side try simplify what they gave.

    [(1+x)(1-x)]^n = (1 - x^2)^n

    Now expand left hand side:
    (nC0 + nC1 x + ... + nCn x^n)(nC0 - nC1 x + nC2 x^2 - ... - nCn x^n)
    Note that the second brackets has nCn x^n as negative as n is odd.
    = [(nC0)^2 - nC0.nC1 x + nC0.nC2 x^2 - .... - nC0.nCn x^n] + [nC1.nC0 x - (nC1)^2 x^2 + ... - nC1.nCn x^(n+1)] + ....
    From subbing in x = 1 and expanding you notice every term will cancel out except for the squares of the coefficients, i.e. = (nC0)^2 - (nC1)^2 + ... + (nCn)^2

    Expand right hand side and sub in x = 1:
    nC0 - nC1 + nC2 - ... - nCn

    Since they're the same equation equate both sides:
    sigma(k=0 to n) (-1)^k (nCk)^2 = nC0 - nC1 + nC2 - .... - nCn
    Via symmetry nC0 = nCn and so on so the right hand side equals 0 if you keep using symmetry and you have your answer

    EDIT: Fixed mistake
    Last edited by pikachu975; 12 Nov 2017 at 12:38 PM.
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    Re: Binomial Theorem Help

    how does (nC0).(nC2)x^2 cancel out?
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    Re: Binomial Theorem Help

    Quote Originally Posted by Andy005 View Post
    how does (nC0).(nC2)x^2 cancel out?
    I think I laid it out wrong. There are multiple terms when expanding, so consider the terms when each term in the LHS is expanded:

    nC0: nC0.nC2 x^2, - nC0.nC(n-2) x^(2n-2)
    nC2: nC2.nC0 x^2, - nC2.nCn x^(n+2)
    nCn: nCn.nC2 x^2, - nCn.nC(n-2) x^(2n-2)

    Via symmetry these all have a coefficient of nC0.nC2 when subbing x = 1 so they cancel out

    My bad it should cancel out after subbing in x = 0, thanks for pointing it out!
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