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Homework Help: Binomial expansions

  1. Aug 17, 2010 #1
    1. The problem statement, all variables and given/known data

    Find, in the simplest form, the coefficient of x^n in the binomial expansion of (1-x)^(-6).

    2. Relevant equations

    3. The attempt at a solution

    i am not sure how to go about with this.
  2. jcsd
  3. Aug 17, 2010 #2
    are u sure you have to find a cofficient contain x^n?

    because u should have a specific value for n, so that you could find the cofficient infront of it, or you should at least have which term you are looking for.
  4. Aug 18, 2010 #3
    Yes, that's the question. Maybe it's asking for the coefficient of for any term in the expression.
  5. Aug 18, 2010 #4


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    Science Advisor

    Since the exponent, -6, is not a positive integer, you will need to use the generalized binomial series:
    [tex](a+ b)^m= \sum_{k=0}^\infty \frac{m(m-1)\cdot\cdot\cdot(m-k+1)}{k!}a^kb^{m-k}[/tex]

    Here, of course, a= 1, b= -x, and m= -6 so this is

    [tex](1- x)^{-6}= \sum_{k=0}^\infty \frac{-6(-7)\cdot\cdot\cdot(-5-k)}{k!}(-1)^kx^{-6-k}[/tex]

    You, apparently, are asked for the coefficient when -6-k= n or when k= -6-n.
  6. Aug 18, 2010 #5

    Is this answer the most simplified?


    the general formula for binomial series for (a+b)^n is different when n is a positive integer and when n is a fractional or negative value?

    [tex](a+ b)^m= \sum_{k=0}^\infty \frac{m(m-1)\cdot\cdot\cdot(m-k+1)}{k!}a^kb^{m-k}[/tex]

    Does it matter if the powers(k and n-k) for a and b is swapped since a+b is commutative?

    This is the continuation of the question:

    Hence, find the coefficient of x^6 and x^7 in (1+2x+3x^2+4x^3+5x^4+6x^5+7x^6)^3
    Last edited: Aug 18, 2010
  7. Aug 19, 2010 #6
    any further hints on this question?
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