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Unformatted text preview: 1.9. COUNTING 59 ÂÃ
nŠ
n
The formula
D
implies (b) when 0 Ä k Ä n. But
k
kŠ.n k /Š
when k is not in this range, neither is n k , so both sides of (b) are zero.
When k is 0, both sides of (c) are equal to 1. When k is negative or
greater than n, both sides of (c) are zero. For 0 < k Ä n, (c) follows from
a combinatorial argument: To choose k elements out of f1; 2; : : : ; ng, we
can either choose n, together with k 1 elements out of f1; 2; : : : ; n 1g,
Â
Ã
n1
which can be done in
ways, or we can choose k elements out of
k1
Â
Ã
n1
f1; 2; : : : ; n 1g, which can be done in
ways.
I
k
ÂÃ
n
Example 1.9.4. The coefﬁcients
have an interpretation in terms of
k
paths. Consider paths in the .x; y/–plane from .0; 0/ to a point .a; b/ with
nonnegative integer coordinates. We admit only paths of a C b “steps,” in
which each step goes one unit to the right or one unit up; that is, each step
is either a horizontal segment from an integer point .x; y/ to .x C 1; y/,
or a vertical segment from .x; y/ to .x; y C 1/. How many such paths are
there? Each path has exactly a steps to the right and b steps up, so a path
can be speciﬁed by a sequence with a R’s and b U’s. Such a sequence
is determined by choosing the positions of the a R’s, so the number of
Â
ÃÂ
Ã
aCb
aCb
sequences (and the number of paths) is
D
.
a
b
ÂÃ
n
The numbers
are called binomial coefﬁcients, because of the folk
lowing proposition:
Proposition 1.9.5. (Binomial theorem). Let x and y be numbers (or variables). For n 0 we have
n
X ÂnÃ
n
.x C y/ D
xk yn k :
k
k D0 Proof. .x C y/n is a sum of 2n monomials, each obtained by choosing x
from some of the n factors, and choosing y from the remaining factors.
For ﬁxed k , the number of monomials x k y n k Â Ã sum is the number of
in the
n
ways of choosing k objects out of n, which is
. Hence the coefﬁcient
k
ÂÃ
n
of x k y n k in the product is
.
I
k ...
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 Fall '08
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 Algebra, Counting

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