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.
$\displaystyle \frac{(5n)!}{(2n)!(3n)!} =$ $\displaystyle \frac{(5n)(5n-1)(5n-2)(5n-3)(5n-4)\cdot\cdot\cdot}{[(2n)(2n-1)(2n-2)(2n-3)(2n-4)\cdot\cdot\cdot][(3n)(3n-1)(3n-2)(3n-3)(3n-4)\cdot\cdot\cdot]}$
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
$\displaystyle \frac{\Gamma(5 n+1)}{\Gamma(2n+1) \times \Gamma(3n+1)}$