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Math Help - Permutation Problem

  1. #1
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    Permutation Problem

    Problem: In how many ways can these letters EFGHIEFGHI be arranged if the same letter musn't appear next to each other?

    since we have 2 E's, F's, G's, H's and I's... The formula should be

    10!
    ------------- = 113400 ways
    2!2!2!2!2!

    Then I combine similar letters [EE]FGHIFGHI for 5 letters


    9!x5
    -------- = 113400 ways
    2!2!2!2!


    That's where i get stumped... need help... XD
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  2. #2
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    Quote Originally Posted by mcroldan08 View Post
    Problem: In how many ways can these letters EFGHIEFGHI be arranged if the same letter musn't appear next to each other?
    You have started it correctly. Now we apply the inclusion/exclusion rule.
    \sum\limits_{k = 0}^5 {\left( { - 1} \right)^k \binom{5}{k}\frac{{\left( {10 - k}\right)!}}{{2^{5 - k} }}}
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  3. #3
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    uhmm... could you walk me through that rule... I'm not familiar with it... XD
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  4. #4
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    Quote Originally Posted by mcroldan08 View Post
    uhmm... could you walk me through that rule... I'm not familiar with it... XD
    If that is true, then I must wonder why you are asked to do this problem?
    Inclusion-Exclusion Principle -- from Wolfram MathWorld
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  5. #5
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    =n(A) + n(B) + n(C) + n(D) + n(E) - n (A&B) - n(A&C) - n(A&D) - ... - n(A&B&C) - n(B&C&D) - ... - n(A&B&C&D) - n(A&B&C&E) - ... + n(A and B and C and D and E)

    Is this right?? XD
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  6. #6
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    Quote Originally Posted by mcroldan08 View Post
    =n(A) + n(B) + n(C) + n(D) + n(E) - n (A&B) - n(A&C) - n(A&D) - ... - n(A&B&C) - n(B&C&D) - ... - n(A&B&C&D) - n(A&B&C&E) - ... + n(A and B and C and D and E)

    Is this right?? XD
    That is the basic idea.
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  7. #7
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    Got it...

    Final Check...

    = 5x9!/2!2!2!2! - 10x(8!/2!2!2!) - 10x(7!/2!2!) - 5x(6!/2!) + 5!
    =48720

    so 113400-48720 = 64680...

    anything I messed up??


    btw... A heap of thanks!! really really appreciated the help...
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  8. #8
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    Quote Originally Posted by mcroldan08 View Post
    Got it...

    Final Check...

    = 5x9!/2!2!2!2! - 10x(8!/2!2!2!) - 10x(7!/2!2!) - 5x(6!/2!) + 5!

    anything I messed up??
    You missed a sign. It should be +.
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  9. #9
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    Ok.. Got it!! Thanks a lot!!
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