# Dice games, part 3

Calculus Level pending

Let $$P_n$$ be the probability that after rolling $$n$$ fair $$n$$-sided dice, one of each natural number $$1$$ through $$n$$ is shown.

Let $$\displaystyle Y = \lim_{n \to \infty} \ln\left(\frac{P_n}{\sqrt{n}}\right)+n$$.

Compute $$\lfloor Y \times 10^9\rfloor$$.

×