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Elementary Combinatorics Q

  1. May 15, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose that a teacher wishes to distribute 25 identical pencils to Ahmed, Bar-
    bara, Carlos, and Dieter such that Ahmed and Dieter receive at least one pencil
    each, Carlos receives no more than five pencils, and Barbara receives at least four
    pencils. In how many ways can such a distribution be made?

    Or, in other words, find integer solutions to [tex] x_1 + x_2 +x_3+x_4=25, x_1>0, x_2>0, x_3\le5, x_4\ge4 [/tex]

    Please let me know if i made any silly errors, but I'm more concerned that I made a fundamental error in the logic of this problem. Thanks!



    The first inequality is
    2. Relevant equations

    The number of integer solutions to the equation [tex] x_1 + x_2 + x_3 \ldots x_n = C, x_i>0 [/tex] is [tex] C-1\choose n-1 [/tex].



    3. The attempt at a solution

    EDIT: got the solution
     
    Last edited: May 16, 2009
  2. jcsd
  3. May 15, 2009 #2
    I would use generating functions.

    Expand (x+x^2+...+x^25)*(x+x^2+...+x^25)*(1+x+x^2+x^3+x^4+x^5)*(x^4+x^5+...x^25) and find the coefficient of x^25.

    I'll do it on Maple and see what I get.
     
  4. May 15, 2009 #3
    I'm getting 980.
     
  5. May 15, 2009 #4
    looking it over again my attempt at a solution is all backwards =/
     
  6. May 16, 2009 #5
    Tada! I'm getting 980 after looking it over as well, don't know what I was thinking when I first attempted the solution. Thanks for your responses, I'll post my solution when I have more time.
     
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