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Math Help - Homework help

  1. #1
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    Homework help

    Find the number of integer solutions of

    1) x1 + x2 +x3 + x4= 15
    x1 >1, x2 >0, x3 > 1, x4> 2


    2) x1 + x2 + x3 + x4 =12
    xi >0, i= 1,2,3,4.
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  2. #2
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    Hello, Laura!

    Here's the second one . . .the easier of the two.


    Find the number of integer solutions of: . x_1 + x_2 + x_3 + x_4 \:=\:12\qquad x_1,x_2,x_3,x_4 \:>\:0

    Consider 12 objects in a row; leave a space between them.

    . . o\:\_\:o\:\_\:o\:\_\:o\:\_\:o\:\_\:o\:\_\:o\:\_\:o  \:\_\:o\:\_\:o\:\_\:o\:\_\:o

    There are 11 spaces.
    We choose 3 of them to insert "dividers."

    So that: . o\:\_\:o\:|\:o\:\_\:o\:\_\:o\:\_\:o \:|\:o\:|\:o\:\_\:o\:\_\:o\:\_\:o\:\_\:o\quad\text  { means: }2 + 4 + 1 + 5

    . . And: . o\:|\:o\:\_\:o\:|\:o\:\_\:o\:\_\:o\:\_\:o\:\_\:o\:  \_\:o\:\_\:o\:|\:o\:\_\:o \quad \text{ means: }1 + 2 + 7 + 2


    Therefore, there are: . {11\choose3} \:=\:165 solutions.

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