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Math Help - NCr (nCk) find binomial coefficient of x^14 for expansion of (3x - 1/3)^16?

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    Post NCr (nCk) find binomial coefficient of x^14 for expansion of (3x - 1/3)^16?

    Going through past paper for the above question to find the coefficient of the above expansion & have seen a guide to using my Casio to work out the value of coefficient at x^14 but it errors for this calculation.
    n should be bigger than r, shouldn't it?
    i.e. n=14 & r=16
    nCr = n!/(n-r)!r! which is 14! / -2! 16! so no suprise there, perhaps the indeces are wrong?

    So, next example seems more likely; for the coefficient of x^18 for expansion of (1/14 x^2 - 7)^16.
    so n=18 & r=16

    nCr = 18! / -2! 16! = 153

    I have a feeling there is something else, but not sure what it is.
    Could someone point in the right direction
    Last edited by froodles01; January 11th 2012 at 03:37 AM.
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    Re: NCr (nCk) find binomial coefficient of x^14 for expansion of (3x - 1/3)^16?

    Quote Originally Posted by froodles01 View Post
    Going the value of coefficient at x^14 but it errors for this calculation. n should be bigger than r, shouldn't it?
    i.e. n=14 & r=16
    nCr = n!/(n-r)!r! which is 14! / -2! 16! so no suprise there, perhaps the indeces are wrong?
    You want x^{14}. So \binom{16}{2}(3x)^{14}\left(\frac{-1}{3}\right)^2
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    Re: NCr (nCk) find binomial coefficient of x^14 for expansion of (3x - 1/3)^16?

    Thank you, the 3x term is raised to 14 for the coefficient of that term, but I don't understand why the 1/3 term is squared not raised to 14 too.

    Can you apply this to the second example, too.
    Thank you.
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    Re: NCr (nCk) find binomial coefficient of x^14 for expansion of (3x - 1/3)^16?

    Quote Originally Posted by froodles01 View Post
    Thank you, the 3x term is raised to 14 for the coefficient of that term, but I don't understand why the 1/3 term is squared not raised to 14 too.
    The terms of the expansion (x+y)^N are \binom{N}{k}x^ky^{N-k}.
    In your question x=3x,~y=\tfrac{-3}{x}~\&~N=16 .
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    Junior Member froodles01's Avatar
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    Re: NCr (nCk) find binomial coefficient of x^14 for expansion of (3x - 1/3)^16?

    Thank you. I'm getting there, but in the second example where N=16, the k value is -2

    Should this look like: (16 -2) (1/4 x^2)^18 (-7)^-2
    = 4.322exp-24
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    Re: NCr (nCk) find binomial coefficient of x^14 for expansion of (3x - 1/3)^16?

    Quote Originally Posted by froodles01 View Post
    So, next example seems more likely; for the coefficient of x^18 for expansion of (1/14 x^2 - 7)^16.
    In this one, N=16,~k=9~\&~(N-k)=7.
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