Unformatted text preview: 60 1. ALGEBRAIC THEMES Corollary 1.9.6.
n
X ÂnÃ
n
(a) 2 D
.
k
k D0
ÂÃ
n
X
kn
(b) 0 D
. 1/
.
k
k D0
n
n
X ÂnÃ
X ÂnÃ
n1
(c) 2
D
D
.
k
k
kD0
kD0
k odd
k even Proof. Part (a) follows from the combinatorial interpretation of the two
sides: The total number of subsets of an n element set is the sum over
k of the number of subsets with k elements. Part (a) also follows from
the binomial theorem by putting x D y D 1. Part (b) follows from the
binomial theorem by putting x D 1; y D 1. The two sums in part (c) are
equal by part (b), and they add up to 2n by part (a); hence each is equal to
2n 1 .
I
Example 1.9.7. We can obtain many identities for the binomial coefﬁcients by starting with the special case of the binomial theorem:
n
X ÂnÃ
n
.1 C x/ D
xk ;
k
k D0 regarding both sides as functions in a real variable x , manipulating the
functions (for example, by differentiating, integrating, multiplying by x ,
etc.), and ﬁnally evaluating for a speciﬁc value of x . For example, differentiating the basic formula gives
n
X ÂnÃ
n.1 C x/n 1 D
k
xk 1:
k
k D1 Evaluating at x D 1 gives
n
X ÂnÃ
D
k
;
k n1 n2 k D1 while evaluating at x D 1 gives
0D n
X . 1/ k D1 k1 ÂÃ
n
k
:
k ...
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 Fall '08
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 Algebra, Binomial Theorem, Sets

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