The Answer May Surprise You

Let \(n\) be a natural number. Let \((a_k)_{k=1}^{n}\) be a permutation of the numbers \(\{1,2,3, ... n\}\). Let \(p_n\) be the probability that \(a_k \neq k\) for all \(1 \le k \le n\).

Find \(\lim_{n \to \infty} \ln(p_n)\).

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