Let \(d_n\) be the number of derangements of \(n\) objects. For example, using the techniques of this section, we find
We can use the formula for \({n \choose k}\) to write this all in terms of factorials. After simplifying, for \(d_3\) we would get
Generalize this to find a nicer formula for \(d_n\text{.}\) Bonus: For large \(n\text{,}\) approximately what fraction of all permutations are derangements? Use your knowledge of Taylor series from calculus.
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