Number of solutions to an equation

  • Context: Undergrad 
  • Thread starter Thread starter praharmitra
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Discussion Overview

The discussion revolves around the integer equation x_1 + x_2 + ... + x_n = m, where x_i are non-negative integers. Participants explore the number of solutions to this equation and the potential for a one-to-one correspondence between solutions and selections of objects, considering combinatorial interpretations.

Discussion Character

  • Exploratory, Mathematical reasoning

Main Points Raised

  • One participant presents the equation and suggests that the number of solutions can be expressed as ^{n+m-1}C_m, proposing a combinatorial interpretation involving selections of objects.
  • Another participant questions whether the selection should involve n-1 objects instead, indicating a potential alternative perspective.
  • A third participant shares a link to the "Stars and Bars" theorem, which may provide relevant insights into the problem.
  • A later reply indicates that the original poster has found a one-to-one correspondence related to the provided link and offers to explain further if needed.

Areas of Agreement / Disagreement

The discussion includes multiple viewpoints regarding the interpretation of the equation and the nature of the correspondence, with no consensus reached on the best approach or interpretation.

Contextual Notes

Participants have not fully resolved the implications of the proposed interpretations or the specifics of the correspondence, leaving some assumptions and definitions open to further clarification.

praharmitra
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Given the following integer equation [tex]x_1 + x_2 + ... x_n = m[/tex] where [tex]x_i \geq 0[/tex] and [tex]x_i[/tex] is an integer for all i.

The number of solutions to the above equation is [tex]^{n+m-1}C_m[/tex]

I was wondering if we could view this as a selection of [tex]m[/tex] objects from a selection of [tex]n + m - 1[/tex] objects.

Is there a 1-to-1 correspondence between a particular solution of the equation, and a particular selection of m objects from a selection of some n + m - 1 objects.

I hope I have made myself clear. I have tried to figure out such a correspondence, but in vain.
 
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How about a selection of n-1 objects?
 
Hey, sorry I haven't replied back on my own thread. Was away.

Anyway, I figured out the required one-to-one correspondence myself. Its pretty much related to the link that @awkward has provided.

So, if anyone still has trouble understanding, I would be glad to explain.

Thanks a lot guys for your help.
 

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