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Math Help - [SOLVED] I have a problem.....

  1. #1
    spank
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    Exclamation [SOLVED] I have a problem.....

    It's a fairly straight forward one i think, but its been a long time since a studied maths!

    The problem is...

    There are 6 slots in a row.

    One tab can be inserted into each slot.

    There are 6 different coded tabs (e.g. 1, 2, 3, 4, 5, 6)

    Therefore the total number of cominations is 6^6.

    However, three or more tabs of the same code cannot lie next to each other.
    i.e. the following combinations cannot exist:
    2222222
    655555
    651113
    144442
    etc..... because there are 3 or more tabs of the same code next to each other.

    What is the total number of allowable combinations?
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  2. #2
    MHF Contributor red_dog's Avatar
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    For the first slot we have 6 possibilities.
    For the second slot we have 5 possibilities (we can't put a tab with the same number as in the first position).
    For the third slot we have 5 possibilities (we can\t put a tab with the same number as in the second slot)
    ..........................................
    For the sixth slot we have 5 possibilities.
    So, using the product rule of combinatorics, we have 6\cdot5\cdot5\cdot5\cdot5\cdot5=6\cdot5^5 possibilities.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by red_dog View Post
    For the first slot we have 6 possibilities.
    For the second slot we have 5 possibilities (we can't put a tab with the same number as in the first position).
    You can have two the same adjacent, it is three that is forbidden.

    For the third slot we have 5 possibilities (we can\t put a tab with the same number as in the second slot)
    ..........................................
    For the sixth slot we have 5 possibilities.
    So, using the product rule of combinatorics, we have 6\cdot5\cdot5\cdot5\cdot5\cdot5=6\cdot5^5 possibilities.
    RonL
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  4. #4
    MHF Contributor red_dog's Avatar
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    Sorry...I misunterstood the problem. I'll think about it.
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  5. #5
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    Count the number that look like: XXXYYY that is 30.
    Count the number that look like: XXXYYZ that is \left( 6 \right)\left( 5 \right)\left( 4 \right)\left( {\frac{{4!}}{2}} \right).
    Count the number that look like: XXXYWZ that is \left( 6 \right)\left( 5 \right)\left( 4 \right)\left( 3 \right)\left( {4} \right).
    Thus, there are 2910 strings that have at least one triple.

    Count the number that look like: XXXXYY that is \left( 6 \right)\left( 5 \right)\left( {\frac{{3!}}{{2!}}} \right).
    Count the number that look like: XXXXYZ that is \left( 6 \right)\left( 5 \right)\left( 4 \right)\left( {3!} \right).
    Thus, there are 810 with a string of four.

    There are 60 with a string of five and 6 with a string of six.

    Those are the one we do not want. Add up and subtract from 6^6.
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  6. #6
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    Quote Originally Posted by spank View Post
    It's a fairly straight forward one i think, but its been a long time since a studied maths!

    The problem is...

    There are 6 slots in a row.

    One tab can be inserted into each slot.

    There are 6 different coded tabs (e.g. 1, 2, 3, 4, 5, 6)

    Therefore the total number of cominations is 6^6.

    However, three or more tabs of the same code cannot lie next to each other.
    i.e. the following combinations cannot exist:
    2222222
    655555
    651113
    144442
    etc..... because there are 3 or more tabs of the same code next to each other.

    What is the total number of allowable combinations?
    Hi spank .
    I have a idea .

    Case 1 : 3 XXX tags next to each other.
    -Choosing 1 number : 10 choices.
    -Choosing the tags's position : 4 choices. (XXX___,_XXX__,__XXX_,___XXX)
    -Choosing the other tags : 9*8*7.

    Case 2 : 4 XXXX tags next to each other.
    -Choosing 1 number : 10 choices.
    -Choosing the tags's position : 3 choices.
    -Choosing the other tags : 9*8.

    Case 3 : 5 XXXXX tags next to each other.
    -Choosing 1 number : 10 choices.
    -Choosing the tags' s position : 2 choices.
    -Choosing the other tags : 9.

    Case 4 : 6 tags next to each other.
    -Choosing a number : 10.

    So , the result is : 6^6 - (10*4*9*8*7 +10*3*9*8 +10*2*9 + 10)
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