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Math Help - Counting...

  1. #1
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    Post Counting...

    Well, hi.

    I've got a problem, I've been trying to solve it but every formula I try to make fails...

    Perhaps this is foolish and I'm not applying correctly the Permutations, Combinations and Factorials...but I don't get to the answer.

    The problem is:

    Let's Suppose that I'm a teacher, I take some notebooks from nine students, and I give them back. ¿How many ways are there to give back the notebooks in a way that no one receives his own notebook?

    What I do, naturally, is to reduce the problem, and try making a formula.

    I look what happens with four notebooks...I started looking how many ways there are...and I find there are 9 ways. With 3 notebooks there are only 2 ways. And with 2 there is only 1. No easy-looking pattern, so I can't make a formula that easy...

    Then I tried using a formula that I already know...the total quantity of ways to give back the notebooks is n!. Then the formula must be
    Counting...-n.png.

    When I try to use all of that I get a weird +...-... thing like n!-n!+(n-1)!-...

    But it's not exactly the formula...and I'm not sure if it really can be done like that...
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  2. #2
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    Re: Counting...

    Quote Originally Posted by Leviathantheesper View Post
    The problem is:
    [FONT=arial black]
    Let's Suppose that [FONT=arial]I'm a teacher, I take some notebooks from nine students, and I give them back. ¿How many ways are there to give back the notebooks in a way that no one receives his own notebook?
    What I do, naturally, is to reduce the problem, and try making a formula.
    I look what happens with four notebooks...I started looking how many ways there are...and I find there are 9 ways. With 3 notebooks there are only 2 ways. And with 2 there is only 1. No easy-looking pattern, so I can't make a formula that easy...

    This is known as the problem of derangement.
    I don't like the notation there, but it gives you the ideas.

    I like the formula: D(n) = (n!)\sum\limits_{k = 0}^n {\frac{{{{( - 1)}^{k }}}}{{k!}}}  \approx \frac{{n!}}{e}.
    Last edited by Plato; May 4th 2013 at 01:04 PM.
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