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Combinatorics Problem

  1. Mar 23, 2010 #1
    1. The problem statement, all variables and given/known data

    Find the coefficient of [tex]x^{9}[/tex] [tex]y^{10}[/tex] in (3[tex]x^{3}[/tex] - 4[tex]y^{2}[/tex])[tex]^{8}[/tex]

    2. Relevant equations

    The professor gave us a somewhat algebraic tactic or shortcut for solving these kinds of problems, mainly consisting of solving for each exponent. It can be somewhat tricky for me to explain with typing, but...

    3. The attempt at a solution

    As per her suggested method, this is what I obtained:

    C(8, k) (3[tex]x^{3}[/tex])[tex]^{8-k}[/tex] * ([tex]-4^{2}[/tex])[tex]^{k}[/tex]

    3(8 - k) = 9, 2k = 10. Thus k = 5 and 8 - k = 3.

    This gives me the answer C(8,5) * [tex]3^{3}[/tex] * [tex](-4)^{5}[/tex].

    But her answer key in the provided review paper claims the answer is actually

    C(8,3) * [tex]3^{3}[/tex] * [tex](-4)^{5}[/tex]

    Have I done something wrong or is her answer incorrect? My text doesn't support her method and she is currently unavailable to assist me.
    Last edited: Mar 23, 2010
  2. jcsd
  3. Mar 23, 2010 #2


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    Homework Helper

    C(8,3)=C(8,5). If you swap k and 8-k you get the key answer. They are both equal and both right.
  4. Mar 24, 2010 #3
    Checked it out, you are correct. Thank you for your assistance.
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