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Number of objects distributed between four people

  1. Dec 8, 2015 #1
    1. The problem statement, all variables and given/known data
    Four people are dealing the total amount of money, which is [itex]1000[/itex] monetary units in terms of [itex]100[/itex] monetary units. Count the number of ways for this distribution if:
    [itex]1)[/itex] Every person doesn't have to get any money
    [itex]2)[/itex] Every person will get at least [itex]100[/itex] monetary units
    [itex]3)[/itex] First person will get at least [itex]500[/itex] m.u. and other three people at least [itex]100[/itex] m.u.

    2. Relevant equations
    -Combinatorics

    3. The attempt at a solution
    The problem doesn't state what is the maximum amount of money that each person can get.
    Assuming that, in [itex]1)[/itex] every person will get the same amount ([itex]200[/itex] m.u.), the total number of counts would be the coefficient with [itex]x^{1000}[/itex] in [itex](1+x+...+x^{200})^4[/itex].

    In [itex]2)[/itex], assuming that the maximum amount for every person is [itex]200[/itex] m.u, the total number of counts would be the coefficient with [itex]x^{1000}[/itex] in [itex](x^{100}+...+x^{200})^4[/itex].

    In [itex]3)\Rightarrow x^{500}(x^{100}+...+x^{200})^3[/itex]

    What do you think, how to solve this problem?
     
  2. jcsd
  3. Dec 8, 2015 #2

    andrewkirk

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    You can convert (2) and (3) into the same problem as (1) with a different total to be distributed, by doing something non-probabilistic before you start the random part. What might that non-probabilistic step be?


    BTW, your formula ##(1+x+....+x^{200})^4## doesn't need to be have such a huge number of terms. Since everything is done in multiples of 100mu, divide the amounts by 100mu before you start.
     
  4. Dec 8, 2015 #3

    Ray Vickson

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    I found the wording of your problem confusing: if you mean that each person can receive 0 or 100 or 200 or ... (i,e, multiples of 100) then you should say that a bit more clearly. If that is what you mean, it would be much easier to let 100 m.u be a new monetary unit. Now the amount each person can receive is an integer number of these new units and we have a total of 10 new units for distribution.

    Also, to clarify: in (3), do you mean (in these new units) that persons 2--4 get at least 1 in total, or that persons 2--4 will get at least 1 each?
     
  5. Dec 9, 2015 #4

    haruspex

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    That would not be a good assumption. As I read it, the whole amount is to be divvied up, and there's no restriction on who gets what. Essentially, you have ten identical things to put in four (distinct) baskets.
     
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