1.9. COUNTING
65
This sum can be simplified as follows:
D
n
D D
n
X
k
D
0
.
1/
k
n
k
.n
k/Š
D
nŠ
n
X
k
D
0
.
1/
k
1
kŠ
:
Since
1
X
k
D
0
.
1/
k
1
kŠ
is an alternating series with limit
1=e
, we have
j
1=e
n
X
k
D
0
.
1/
k
1
kŠ
j
1=.n
C
1/Š;
so
j
D
n
nŠ=e
j
nŠ=.n
C
1/Š
D
1=.n
C
1/:
Therefore,
D
n
is the integer closest to
nŠ=e
.
Example 1.9.14.
Ten diners leave coats in the wardrobe of a restaurant.
In how many ways can the coats be returned so that no customer gets his
own coat back?
The number of ways in which the coats can be returned,
each to the wrong customer, is the number of derangements of 10 objects,
D
10
D
1; 333; 961
.
The primary goal of our discussion of inclusionexclusion is to obtain
a formula for the
Euler
'
function
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 EVERAGE
 Algebra, Natural Numbers, Counting, Natural number, Prime number

Click to edit the document details