# Simplify factorials help!

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• Apr 8th 2010, 12:17 PM
Frankler
Simplify factorials help!
Please can anyone help me to simplify
$\displaystyle (5n)!/(2n)!(3n)!$

Thanks, this is really bugging me.
• Apr 8th 2010, 12:31 PM
Anonymous1
Quote:

Originally Posted by Frankler
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.

$\displaystyle \frac{(5n)!}{(2n)!(3n)!} =$ $\displaystyle \frac{(5n)(5n-1)(5n-2)(5n-3)(5n-4)\cdot\cdot\cdot}{[(2n)(2n-1)(2n-2)(2n-3)(2n-4)\cdot\cdot\cdot][(3n)(3n-1)(3n-2)(3n-3)(3n-4)\cdot\cdot\cdot]}$
• Apr 8th 2010, 02:09 PM
harish21
Quote:

Originally Posted by Frankler
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

$\displaystyle \frac{\Gamma(5 n+1)}{\Gamma(2n+1) \times \Gamma(3n+1)}$
• Apr 11th 2010, 03:08 AM
Frankler
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 $\displaystyle \frac{\Gamma(5 n+1)}{\Gamma(2n+1) \times \Gamma(3n+1)}$
• Apr 11th 2010, 05:51 AM
Archie Meade
Quote:

Originally Posted by Frankler
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

$\displaystyle \binom{5n}{2n}=(5n)_{C_{(2n)}}=\frac{(5n)!}{(5n-2n)!(2n)!}$
• Apr 11th 2010, 10:08 AM
Frankler
Yes, that where I started with this problem, but does anyone know how it simplifies??
• Apr 12th 2010, 01:57 AM
mr fantastic
Quote:

Originally Posted by Frankler
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?
• Apr 13th 2010, 02:58 AM
Frankler
Just to finish answering a question. Is there no way to simplify it? I'll just leave it then, thanks :)