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Thread: Combinatorics

  1. #1 Combinatorics 
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    Is there a proof that the number of combinations of items will always be ? Just something I've been worrying about for the last few days...


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  3. #2 Re: Combinatorics 
    Forum Ph.D. Leszek Luchowski's Avatar
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    Quote Originally Posted by Liongold
    Is there a proof that the number of combinations of items will always be ? Just something I've been worrying about for the last few days...
    You mean permutations (ordered arrangements). Yes there is, and it's quite simple.

    You have n distinct elements (say n persons) to be placed in n distinct positions (say at the table at some meeting where it does matter who is seated where).

    You start filling the seats from first to last.

    For the first seat, you have n possible canditates to choose from.

    For the second, n-1.

    ----

    For the last but one seat, just 2 candidates

    For the last seat, you are left with 1 person and you have to put him or her there, no more choice.

    Every possible arrangement can be reached in this way.

    Making a different choice at any step will result in a different arrangement.

    The number of possible different sequences of your choices is

    n*(n-1)*...2*1=n!

    Which completes the proof.


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  4. #3  
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    Ok, thanks!
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  5. #4  
    Forum Freshman pbandjay's Avatar
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    I always just thought of it as the awesomeness that is... The Multiplication Principle: running from n to 1.
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