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Unformatted text preview: 65 1.9. COUNTING This sum can be simpliﬁed as follows:
ÂÃ
n
X
kn
Dn D D
. 1/
.n
k k /Š D nŠ k D0 Since 1
X
k D0 . 1/k n
X . 1/k k D0 1
:
kŠ 1
is an alternating series with limit 1=e , we have
kŠ
j1=e n
X . 1/k k D0 1
j Ä 1=.n C 1/Š;
kŠ so
jDn nŠ=e j Ä nŠ=.n C 1/Š D 1=.n C 1/: Therefore, Dn 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,
D10 D 1; 333; 961.
The primary goal of our discussion of inclusionexclusion is to obtain
a formula for the Euler ' function:
Deﬁnition 1.9.15. For each natural number n, '.n/ is deﬁned to be the the
cardinality of the set of natural numbers k < n such that k is relatively
prime to n. Lemma 1.9.16. Let k and n be be natural numbers, with k dividing n. The
number of natural numbers j Ä n such that k divides j is n=k . Proof. Say kd D n. The set of natural numbers that are no greater than
n and divisible by k is fk; 2k; 3k; : : : ; d k D ng, so the size of this set is
d D n=k .
I Corollary 1.9.17. If p is a prime number, then for all k
p k 1 .p 1/. 1, '.p k / D ...
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 Fall '08
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 Algebra, Counting

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