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{Combinatorics} Coins distributed among people.

  1. Jul 30, 2015 #1
    1. The problem statement, all variables and given/known data
    Find the number of ways to distribute 55 identical coins among three people, so that everyone gets an odd number of coins.



    2. Relevant equations
    Stars and Bars Formula



    3. The attempt at a solution

    (n+r-1,n-1)

    Ways to place r indistinguishable objects into n distinguishable boxes.
    C(57,2)=1596 total ways



    Thats about it. If it was even I could use same formula I think but with groups of objects instead. Since its odd I'm unsure.


    My weak guess was I took 55/3 and used the same formula and got roughly 196 total ways I'm sure that is wrong though. Thanks for any help.
     
    Last edited: Jul 30, 2015
  2. jcsd
  3. Jul 30, 2015 #2
    My new attempt which came out right

    My attempt thinking of it as solutions x1+x2+x3=55 solutions to the each such that each is odd z1+z2+z3=28 with no restrictions is the same as y1+y2+y3=52 such that each solution is even

    Thus x1=y1+1=2z1+1

    So number of ways will be 378
     
  4. Jul 30, 2015 #3
    I agree with your answer but don't follow your reasoning !
    Mine is that if you've got ##p_1,p_2,p_3## coins in each box, all these numbers being odd, you can write ## p_i = 2q_i - 1, \ q_i \ge 1 ##. Therefore your problem is equivalent to ## q_1 + q_2 + q_3 = 29 ## which is a classic situation of 'stars and bars' theorem.
     
  5. Jul 31, 2015 #4

    haruspex

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    You mean =26, right? The 2 gets added later.
     
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