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{Number theory} Integer solutions

  1. Aug 16, 2015 #1
    1. The problem statement, all variables and given/known data
    ##x_1+x_2 \cdots x_{251}=708## has a certain # of solutions in positive integers ##x_1 \cdots x_{251}##
    Now the equation ##y_1+y_2 \cdots y_{n}=708## also has the same number of positive integer solutions ##y_1, \cdots y_n## Where ##n \neq251## What is ##n##

    2. Relevant equations
    I think this is a stars and bars problem but I'm not super familiar with it still

    3. The attempt at a solution
    So looking at the stars and bars page it seems that ##{m \choose k}={m \choose m-k}## so then would ##n## in this case just be ##457##? In this case ##m=707## and ##k=250##

    Edit figured it out it is ##458## I just forgot to add 1 back to the original ##n##
     
    Last edited: Aug 16, 2015
  2. jcsd
  3. Aug 16, 2015 #2

    HallsofIvy

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    I don't think I understand the notation here. You say this problem "has a certain # of solutions in positive integers x 1 ⋯x251" I would take that to mean that it has 251 solutions. You then say "y1+y2 \cdots yn=708 also has the same number of positive integer solutions y1 ,⋯yn y_1, \cdots y_n Where n≠251".

    If the first equation has 251 solutions and the next has "the same number" how is it not 251?

    And I have no idea what a "stars and bars problem" and a "stars and bars page" are!
     
  4. Aug 16, 2015 #3

    haruspex

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    I believe it refers to the problem of how many ways of placing r identical objects into n distinct buckets. Maybe the 'bars' represent the divisions between the buckets.
    The posted solution, after correction, looks right.
     
  5. Aug 16, 2015 #4
    No, the OP simply means that it "has j solutions", where j is unknown.

    This refers to an interesting class of problems with an elegant path to solution; there are pages that explain this further on both Mathworld and Wikipedia which can be found using a well-known search engine.

    Is the question asking for the number of unique solutions, or are permutations of ## x_i ## permitted?
     
  6. Aug 16, 2015 #5
    ... and have you dealt properly with solutions ending with a bar (that will not sum to 708) and solutions starting with one or more stars that will not satisfy that ## x_i ## are positive?

    Well done for spotting the stars and bars analogue though, it is not immediately obvious.
     
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