Solution
Derangements
Show the problem again
n letters are placed randomly into n addressed envelopes, one per envelope. As n approaches infinity, what does the probability that no letter ends up in its correct envelope converge to? Express as a fraction or decimal rounded to 4 decimal places.
Worked solution
The probability converges to 1/e ≈ 0.3679. The number of derangements of n items is n!(1 − 1/1! + 1/2! − 1/3! + ... + (−1)^n/n!), and dividing by n! gives the probability. As n → ∞, this sum converges to e^(−1) = 1/e by the Taylor series for e^x at x = −1.
Source: The 'probleme des rencontres', posed by Pierre Remond de Montmort (1708). Statement written for AxiomIQ.