How to solve $x(x-1)(x-2)(x-3)(x-4)(x-5)....(x-999)$?
I'm calculating the probability of $1000$ multicast group having different address. Multicast address space is $2^{28}$.
So I tried to calculate it like this.
$$(2^{28} / 2^{28}) ((2^{28} - 1)/ 2^{28}) ... ((2^{28} - 999)/ 2^{28})$$
$$2^{28} (2^{28} - 1)(2^{28} - 2)... (2^{28} - 999) / (2^{28})^{1000}$$
Substitute $2^{28}$ with $x$,
$$x(x-1)(x-2)...(x-999) / x^{1000}$$
So, where do I go from there?
I just have no idea how to simplify the numerator
You can get a good approximation by using the large-$x$ asymptotic expansion of the ratio of two gamma functions. In particular, $$ \frac{{x(x - 1)(x - 2) \cdots (x - 999)}}{{x^{1000} }} = \frac{{\Gamma (x + 1)}}{{x^{1000} \Gamma (x - 999)}} \approx 1 - \frac{{{\rm 499500}}}{x} + \frac{{{\rm 124583708250}}}{{x^2 }}. $$ Use this with your large $x=2^{28}$.