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Math Help - find the number of different order triples

  1. #1
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    find the number of different order triples

    Find the number of different order triples (a,b,c) s.t a+b+c=12 if

    a) a,b,c must be nonegetive integers

    b) a,b,c must be positive integers


    is this permatutaion or combination? Suppose there is 12 choices for a , then b will be depend on a, and c will depend on both a and b
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  2. #2
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    Re: find the number of different order triples

    Hello, wopashui!

    Believe it or not this is a Combination problem.
    But you'll have see the explanation.


    Find the number of different ordered triples (a,b,c)
    such that a+b+c\,=\,12 if:

    . . a) a,b,c must be nonnegetive integers.

    Place 12 objects in a row, inserting a space before, after and between them.

    . . \_\;*\;\_\;*\;\_\;*\;\_\;*\;\_\;*\;\_\;*\;\_\;* \;\_\;*\;\_\;*\;\_\;*\;\_\;*\;\_\;*\;\_


    Place two "dividers" in any of the 13 spaces.

    So that: . *\;*\;*\;*\;*\,|\,*\;*\;*\;*\,|\,*\;*\;\:* .represents: 5 + 4 + 3

    . . and: . *\;*\;*\;*\,||\,*\;*\;*\;*\;*\;*\;*\;\,* .represents: 4 + 0 + 8


    If the dividers are placed in two different spaces, there are: . _{13}C_2 = 78 ways.

    If the dividers are placed in the same space, there are: 13 ways.

    Hence, there are: . 78 + 13 \,=\,91 ways to place the dividers.

    Therefore, there are 91 triples of nonnegative integers whose sum is 12.




    b) a,b,c must be positive integers.

    Place 12 objects in a row, inserting a space between them.

    . . *\;\_\;*\;\_\;*\;\_\;*\;\_\;*\;\_\;*\;\_\;* \;\_\;*\;\_\;*\;\_\;*\;\_\;*\;\_\;*


    Choose 2 of the 11 spaces and insert "dividers".

    So that: . *\;*\;*\;*\;*\,|\,*\;*\;*\;*\,|\,*\;*\;\:* .represents: 5 + 4 + 3

    . . and: . *\;*\;*\;*\,|\,*\;*\;*\;*\;*\;*\;*\,|\,* .represents: 4 + 7 + 1


    The dividers are placed in two different spaces.
    . . There are: . _{11}C_2 = 55 ways.

    Therefore, there are 55 ordered triples of positive integers whose sum is 12.

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    Re: find the number of different order triples

    thanks alot, but i dun understand why is the second part only has 11C2, not 13C2?
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    Re: find the number of different order triples

    anyone can explain?
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    Re: find the number of different order triples

    Quote Originally Posted by wopashui View Post
    Find the number of different order triples (a,b,c) s.t a+b+c=12 if
    b) a,b,c must be positive integers
    Quote Originally Posted by wopashui View Post
    thanks alot, but i dun understand why is the second part only has 11C2, not 13C2?
    Think of the 12 as being twelve 1's. Lets make the variables positive by putting a 1 is each to begin with. Now that leaves us with nine 1's to distribute: \binom{9+3-1}{9}.
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    Re: find the number of different order triples

    thx,, so the first part would be 12+3-1 C 14, which is 91
    Last edited by wopashui; October 9th 2011 at 02:56 PM.
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    Re: find the number of different order triples

    Quote Originally Posted by wopashui View Post
    thx,, so the first part would be 12+3-1 C 14, which is 91
    @wopashui,
    You have absolutely no idea about any of this.
    You are either playing us for fools or you just don't get it.
    I am done with you in any case.
    Good luck.
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