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 \(\lfloor Y \times 10^9\rfloor\).

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