We have a deck with $n$ cards enumerated $1,2,\ldots,n$. The deck is shuffled. What is the probability of exactly one card to remain on its original position? What is the limit as $n$ rises to infinity?
$$ \begin{array}{rcl} \{1\} & : & \dfrac 11 \\[6pt] \{12,21\} & : & \dfrac 02 \\[6pt] \{123,132,213,231,312,321\} & : & \dfrac 36 \\[6pt] & \vdots \end{array} $$
At $n = 100 0$ and $10 000$ trials:
$$ \begin{array}{rcl} \text{value of }n & & \text{probability} \\ \hline 1000 & & 0.3739 \\ 1001 & & 0.3689 \\ 1002 & & 0.3722 \\ 1003 & & 0.3638 \\ 1004 & & 0.3707 \\ 1005 & & 0.3664 \\ 1006 & & 0.3616 \\ 1007 & & 0.3728 \\ 1008 & & 0.3702 \\ 1009 & & 0.3801 \end{array} $$
At $n = 100 000$ and $10 000$ trials:
$\text{value of } n \quad \text{probability}$
$\quad 100000 \quad \quad 0.3659$
$\quad 100001 \quad \quad 0.3552$
$\quad 100002 \quad \quad 0.356$
$\quad 100003 \quad \quad 0.367$
$\quad 100004 \quad \quad 0.3738$
$\quad 100005 \quad \quad 0.3647$
$\quad 100006 \quad \quad 0.3654$
$\quad 100007 \quad \quad 0.3637$
$\quad 100008 \quad \quad 0.3718$
$\quad 100009 \quad \quad 0.3708$
Apparently, probability approaches $0.36-0.38$, but how can one derive it analytically?
