Dice games, part 3

$P_n=\frac{n!}{n^n}$

From Stirling's formula:

$n! \sim \sqrt{2\pi n} \cdot \frac{n^n}{e^n}$ $\frac{n!}{n^n} \sim \sqrt{2\pi n} \cdot e^{-n}$ $Y = \lim_{n\to \infty} \; \ln\left(\frac{\frac{n!}{n^n}}{\sqrt{n}}\right) + n = \lim_{n\to \infty} \; \ln\left(\frac{\sqrt{2\pi n}}{\sqrt{n}}\cdot e^{-n} \right) + n$

$= \lim_{n \to \infty} \; \ln(\sqrt{2\pi}) + \ln (e^{-n}) + n = \lim_{n \to \infty} \; \frac{1}{2}\ln(2\pi) - n + n = \frac{1}{2}\ln(2\pi)$

$\left\lfloor \frac{1}{2}\ln(2\pi) \times 10^9\right\rfloor = \boxed{918938533}$