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Unformatted text preview: 58 1. ALGEBRAIC THEMES Then arrange the ﬁrst 2 (in 2Š D 2 ways) and the last n 2 (in .n 2/Š
ways). This strange process for building permutations gives the formula
ÂÃ
n
nŠ D
2Š .n 2/Š :
2
Now dividing by 2Š .n 2/Š gives
ÂÃ
nŠ
n.n 1/
n
D
D
:
2
2Š.n 2/Š
2
The virtue of this argument is that it remains valid if 2 is replaced by
any k , 0 Ä k Ä n. So we have the following:
Proposition 1.9.2. Let n be a natural number and let k be an integer in
ÂÃ
n
the range 0 Ä k Ä n. Let
denote the number of k element subsets of
k
an nelement set. Then
ÂÃ
nŠ
n
:
D
k
kŠ.n k /Š
ÂÃ
n
We extend the deﬁnition by declaring
D 0 if k is negative or
kÂÃ
ÂÃ
0
0
greater than n. Also, we declare
D 1 and
D 0 if k ¤ 0. Note
0
kÂ Ã
ÂÃ ÂÃ
n
n
n
that
D
D 1 for all n 0. The expression
is generally read
0
n
k
as “n choose k .”
ÂÃ
n
Here are some elementary properties of the numbers
.
k
Lemma 1.9.3. Let n be a natural number and k 2 Z.
ÂÃ
n
(a)
is a nonnegative integer.
kÃ Â
Â
Ã
n
n
(b)
D
.
Âk Ã Ân kÃ Â
Ã
n
n1
n1
(c)
D
C
.
k
k
k1 ÂÃ
n
Proof. Part (a) is evident from the deﬁnition of
.
k ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Permutations

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