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Number of solutions to an equation

  1. Jul 13, 2010 #1
    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.
     
  2. jcsd
  3. Jul 13, 2010 #2

    Hurkyl

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    How about a selection of n-1 objects?
     
  4. Jul 14, 2010 #3
  5. Jul 21, 2010 #4
    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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