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Thread: Simplify factorials help!

  1. #1
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    Question Simplify factorials help!

    Please can anyone help me to simplify
    $\displaystyle (5n)!/(2n)!(3n)!$

    Thanks, this is really bugging me.
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  2. #2
    Super Member Anonymous1's Avatar
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    Quote Originally Posted by Frankler View Post
    Please can anyone help me to simplify
    $\displaystyle (5n)!/(2n)!(3n)!$

    Thanks, this is really bugging me.
    I could be wrong about this, but I don't know how much simpler this can get...

    You can expand it out and see what cancels.

    $\displaystyle \frac{(5n)!}{(2n)!(3n)!} =$ $\displaystyle \frac{(5n)(5n-1)(5n-2)(5n-3)(5n-4)\cdot\cdot\cdot}{[(2n)(2n-1)(2n-2)(2n-3)(2n-4)\cdot\cdot\cdot][(3n)(3n-1)(3n-2)(3n-3)(3n-4)\cdot\cdot\cdot]}$
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  3. #3
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by Frankler View Post
    Please can anyone help me to simplify
    $\displaystyle (5n)!/(2n)!(3n)!$

    Thanks, this is really bugging me.
    The (complete) gamma function $\displaystyle \Gamma(n)$ is defined to be an extension of the factorial to complex and real number arguments. It is related to the factorial by

    $\displaystyle \Gamma(n) = (n-1)!$


    So your expression can be stated as

    $\displaystyle \frac{\Gamma(5 n+1)}{\Gamma(2n+1) \times \Gamma(3n+1)}$
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  4. #4
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    Thanks both of you. I've tried cancelling it, and got rid of the (3n)!, but I don't see how that helps...
    And how do you simplify $\displaystyle
    \frac{\Gamma(5 n+1)}{\Gamma(2n+1) \times \Gamma(3n+1)}
    $
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  5. #5
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    Quote Originally Posted by Frankler View Post
    Please can anyone help me to simplify
    $\displaystyle (5n)!/(2n)!(3n)!$

    Thanks, this is really bugging me.
    It's basically
    choose 2n elements from 5n elements

    $\displaystyle \binom{5n}{2n}=(5n)_{C_{(2n)}}=\frac{(5n)!}{(5n-2n)!(2n)!}$
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  6. #6
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    Yes, that where I started with this problem, but does anyone know how it simplifies??
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  7. #7
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    Quote Originally Posted by Frankler View Post
    Yes, that where I started with this problem, but does anyone know how it simplifies??
    Why are you so determined to try and simplify it?
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  8. #8
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    Just to finish answering a question. Is there no way to simplify it? I'll just leave it then, thanks
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