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In a hidden friend, suppose no one takes oneself. We say that the hidden friend has "marmalade" if there are two people

A

and

B

such that A took

B

and

B

took

A

. For each positive integer n, let

f (n)

be the number of hidden friends with n people where there is no “marmalade”, i.e.

f (n)

is equal to the number of permutations

\sigma

of {

1, 2,. . . , n

} such that:
*

\sigma (i) \neq i

for all

i=1,2,...,n

* there are no

1 \leq i <j \leq n

such that

\sigma (i) = j

and

\sigma (j) = i.

Determine the limit

\lim_{n \to + \infty} \frac{f(n)}{n!}

Problem Statement