*Proof*.
For each

we calculate the contribution. If

but

is in no

then there is a contribution

to the left. The
only contribution to the right is

when

. If

and

is non-empty
then the contribution to the right is

, the same as
on the left.

**Example 8.6.17** (Derangements)
Suppose we have

mathematicians at a meeting.
Leaving the meeting they pick up their overcoats at random. In how many
ways can this be done so that none of them has his own overcoat. This
number is

, the number of derangements of

objects.

*Proof*.
[Solution]
Let

be the number of ways in which mathematician number

collects his own
coat. Then

.
If

with

then

. Thus

Thus

is the nearest integer to

, since

as

.