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Math Help - no of solutions to w+x+y+z=8?

  1. #1
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    no of solutions to w+x+y+z=8?

    How many solutions in non-negative integers are
    there to the equation
    w +x+y +z = 8?

    and
    How many solutions in positive integers?

    I think for the first part i am just looking at the number of 8 element multisubsets of a 4 element set e.g (8+4-1) choose 8?
    For the second part I am a bit confused,
    thanks for any help
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  2. #2
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    For the second part think this way.
    Put a one in each variable.
    That leaves four to put into four places.
    \dbinom{4+4-1}{4}.
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  3. #3
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    Ah thank you that makes sense
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  4. #4
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    Hello, hmmmm!

    How many solutions in non-negative integers are
    there to the equation: . w +x+y +z \:=\: 8 ?

    Place 8 objects in a row, leaving a space before, after and between them.
    . . <br />
\_\;\circ\;\_\;\circ\;\_\;\circ\;\_\;\circ\;\_\;\c  irc\;\_\; \circ\;\_\;\circ\;\_\; \circ\;\_


    Place 3 "dividers" in any of the 9 spaces.

    . . \circ \circ | \circ \circ \circ| \circ \circ |\circ\;\text{  represents }\,2 + 3 + 2 + 1

    . . \circ \circ || \circ \circ \circ \circ \circ |\circ\;\text{  represents }\,2 + 0 + 5 + 1

    . . |\circ \circ \circ | \circ \circ \circ \circ \circ|\;\text{  represents }\,0 + 3 + 5 + 0

    . . \circ \circ \circ \circ \circ \circ |||\circ \circ\;\text{  represents }\,6 + 0 + 0 + 2

    . . \circ \circ \circ \circ \circ\circ \circ\circ|||\;\text{  represents }\,8 + 0 + 0 + 0


    For each of the three dividers, there are 9 choices for location.

    There are: . 9^3 \,=\,729 non-negative solutions to the equation.




    How many solutions in positive integers?

    Place 8 objects in a row, leaving a space between them.
    . . \circ\;\_\; \circ\;\_\;\circ\;\_\;\circ\;\_\;\circ\;\_\;\circ\  ;\_\;\circ\;\_\;\circ


    Select three of the 7 spaces in which to insert the dividers.

    . . There are: . _7C_3 \:=\:{7\choose3} \:=\:35 ways.

    There are 35 positive integer solutions to the equation.

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