limit N->infty

["Sere"]

Hi there, i would like to know
how to deduce the expression

(pi N/2)^{-1/2} exp[-2(N/2-n)^2 /N]         (N -> infty)

starting from

N! / ( n! (N-n)! 2^N )

remembering that for great values of n we have
n! ~ n^n e^{-n}.

Thank you, Sere