Probability a permutation has a fixed point

#IE#probability#fixed point

Statement

What is the probability that a random permutation of $n$ elements has at least one fixed point?

Hint

$1-D_n/n!$ .

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Compute. Probability $=1-D_n/n!\to1-1/e\approx0.632$ .

For small $n$ . $n=4$ : $1-9/24=15/24=5/8=0.625$ .

Conclusion. Probability tends to $1-1/e\approx0.632$ .