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Math Help - Ho many integral solutions are there...

  1. #1
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    Ho many integral solutions are there...

    I am asked to find how many integral solutions there..

    x1+x2+x3+..........+x10=27

    It asked ...

    a.if all x>=1

    b.if x1 ≥ 2, x2 ≥ 3, and the remaining xi ≥ 0

    c.if all xi ≥ 1, and x1 ≤ 16

    d.if all xi ≥ 1, and x1 ≤ 3 and x2 ≤ 2


    now for the first one it was easy I think

    26 choose 9

    But once it is >=0 I get a little confused.

    First say it said "all Xi>=0"

    Is the formula "n+k-1 choose k"?


    Then for question b. would I go...

    Y1=X1-1
    Y2=X2-2

    Then +1 for each of the remaining xi which is +8?

    So for b it would be

    27-1-2+8=32

    So...

    b. 32 choose 9?


    For
    c.

    Would it be 26 choose 9 subtract 10 choose 9 (number of solution with x1>16)
    Last edited by ehpoc; November 15th 2011 at 12:36 PM.
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  2. #2
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    Re: Ho many integral solutions are there...

    The x_i\geq 0 is just a condition that needs to be satisfied in order for a finite amount of solutions to exist. If anything other than positive integer solutions were possible among the x values, there would be infinitely many solutions to the equation. Think about it, say the situation in which x_3 = 1 and x_4 = -1 satisfied the equation, then so would x_3 = 2 and x_4 = -2, x_3 = 3 and x_4 = -3 etc.

    b) (27-(2+3)+10-1)\choose (10-1)= {31}\choose{9}
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  3. #3
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    Re: Ho many integral solutions are there...

    Quote Originally Posted by ehpoc View Post
    I am asked to find how many integral solutions there.
    x1+x2+x3+..........+x10=27

    c.if all xi ≥ 1, and x1 ≤ 16

    d.if all xi ≥ 1, and x1 ≤ 3 and x2 ≤ 2
    Is the formula "n+k-1 choose k"?

    For
    c.
    Would it be 26 choose 9 subtract 10 choose 9 (number of solution with x1>16)
    A bit LaTeX makes it easier to read.
    [TEX]\binom{26}{9}[/TEX] gives \binom{26}{9}
    [TEX]x_1+x_2+\cdots+x_{10}=27[/TEX] gives x_1+x_2+\cdots+x_{10}=27.

    For c) note that there is only ten cases where x_1>16 if x_i\ge 1. Why?
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