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Math Help - Coefficient in the expansion

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    Coefficient in the expansion

    What is the coefficient of x^6 in the expansion (1+x+x^2)^9. Explain please
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    Quote Originally Posted by ranyeng View Post
    What is the coefficient of x^6 in the expansion (1+x+x^2)^9. Explain please
    Count the number of ways to rearrange each of these string.
    111xxxxxx,~1111xxxxx^2,~ 11111xxx^2x^2,~ \&~111111x^2x^2x^2
    For example: the string 1111xxxxx^2 can be rearranged in \frac{9!}{(1!)(4!)(4!)} ways.
    Note that product is x^6.
    The sum of the numbers of rearrangements is the coefficient of x^6.
    Last edited by Plato; April 6th 2010 at 01:59 PM.
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    Can you please explain with the above example( better if you write the steps)
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    Quote Originally Posted by ranyeng View Post
    Can you please explain with the above example( better if you write the steps)
    No sorry that happens to be your job.

    I will say the number of ways to rearrange the string aabbbcccc is \frac{9!}{(2!)(3!)(4!)}.
    If you do not understand that rule, then you have no business trying this question.
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    Quote Originally Posted by ranyeng View Post
    What is the coefficient of x^6 in the expansion (1+x+x^2)^9. Explain please
    Another approach is to make use of the identity

    (1+x+x^2)^9 = \left( \frac{1-x^3}{1-x} \right) ^9 = (1-x^3)^9 \; (1-x)^{-9}

    then expand (1-x^3)^9 and  (1-x)^{-9} by the binomial theorem.

    (Assuming you know how to use the binomial theorem for negative exponents.)
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