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Unformatted text preview: he LHS, and the second way
of counting (using the sum rule) yields the RHS of the following
formula: n+1
k =
n
n
+
k−1
k 36 Vandermonde’s Identity
Let m, n, and r be nonnegative integers, r ≤ min(n, m).
Then
r
m+n
m
n
=
.
r
r−k
k
k=0 In particular, when choosing m = n = r, we get
2n
n = n
k=0 37 n
k 2 . Vandermonde’s Identity (2)
Proof: We will prove this by counting in two diﬀerent
ways. Let S and T be two disjoint sets with m = S 
and and n = T .
Counting in the ﬁrst way:
n+m
We can choose r elements from S ∪ T in r ways. Counting in the second way: We can pick r elements
from S ∪ T by picking r − k elements from S and k
elements from T , where 0≤ k . By the product
≤r
m
n
rule, this can be done in r−k k ways. Hence the
total number of ways to pick r elements from S ∪ T
is
r
m
n
.
r−k
k
k=0 38 Permutations and Combinations
with Repetitions 39 Motivation
So far, we assumed that (a) the elements are clearly
distinguishable and (b) each element is chosen at most once
for a permutation and combinations.
We will still keep the assumption (a) that the elements
from which we choose are clearly distinguishable. However,
we will now assume (b’) that each element can be chosen
repeatedly in permutations and combinations. 40 Multisets
A multiset is a generalization of the notion of a
set in which elements are allowed to appear more
than once.
For example, {{a,a,b,c,c,d,d,d}} contains the elements
 a with multiplicity 2,
 b with multiplicity 1,
 c with multiplicity 2,
 d with multiplicity 3.
41 rCombinations with Repetition
Let S be a set with n elements. An rcombination
with repetition of S is a multisubset with r elements
of the set S .
The number of rcombinations with repetitions
of a set S with n elements is denoted by
n
.
r 42 rCombinations with Repetition
Theorem. The number of rcombination with repetition of a set with n elements is given by
n+r−1
n
.
=
r
r...
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 Fall '11
 math

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