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Discrete Mathematics
and Its Applications
Sixth Edition
By Kenneth Rosen
Chapter 5
Counting
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5.1 The Basics of Counting
5.2 The Pigeonhole Principle
5.3 Permutations and
Combinations
5.4 Binomial Permutations and
Combinations
5.5 Generalized Permutations and
Combinations
5.6 Generating Permutations and
Combinations
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5.1 The Basics of Counting
• Basic Counting Principles
– The product rule: Suppose that a procedure can be
broken down into a sequence of two tasks. If there are
n1 ways to do the first task and for each of these ways,
there are n2 ways to do the second task, then there
are n1n2 ways to do the procedure
• Ex.1-10
– The sum rule: If a task can be done either in one of n1
ways or in one of n2 ways, where none of the set of n1
ways is the same as any of the set of n2 ways, then
there are n1+n2 ways to do the task
• Ex. 11-13
• The product rule: If A1,A2 ,…,Am are finite sets,
then the number of elements in the Cartesian
product of these sets is the product of the
number of elements in each set.
– |A1 A2  … Am|= |A1||A2| …|Am|
• The sum rule: If A1,A2 ,…,Am are disjoint finite
sets, then the number of elements in the union of
these sets is the sum of the number of elements
in each set.
– |A1 A2  …  Am|= |A1|+|A2|+ …+|Am|
More Complex Counting Problems
• Ex. 14-16
FIGURE 1 (5.1)
FIGURE 1 Internet Addresses (IPv4).
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The Inclusion-Exclusion Principle
• Suppose a task can be done in n1 or n2
ways, but some of the set of n1 ways are
the same as some of the n2 ways, we have
to subtract the number of ways to do the
task that is both among the set of n1 ways
and the set of n2 ways
– The subtraction principle
– |A1 A2 |= |A1|+|A2|- |A1 A2 |
– Ex. 17-18
Tree Diagrams
• Counting problems can be solved using
tree diagrams
– Leaves: possible outcomes
– Ex. 19-21
FIGURE 2 (5.1)
FIGURE 2 Bit Strings of Length Four without Consecutive 1s.
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FIGURE 3 (5.1)
FIGURE 3 Best Three Games Out of Five Playoffs.
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FIGURE 4 (5.1)
FIGURE 4 Counting Varieties of T-Shirts.
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5.2 The Pigeonhole Principle
• If there are more pigeons than pigeonholes, then
there must be at least one pigeonhole with at
least two pigeons in it
• Theorem 1: (The Pigeonhole Principle) If k is a
positive integer and k+1 or more objects are
placed into k boxes, then there is at least one box
containing two or more of the objects.
– Also called the Dirichlet drawer principle
– Proof (by contraposition)
FIGURE 1 (5.2)
FIGURE 1 There Are More Pigeons Than Pigeonholes.
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• Corollary 1: A function from a set with
k+1 or more elements to a set with k
elements is not one-to-one.
– proof
• Ex.1-4
The Generalized Pigeonhole Principle
• Theorem 2: (The Generalized Pigeonhole
Principle) If N objects are placed into k
boxes, then there is at least one box
containing at least N/k objects.
– Proof
– Ex. 5-8
Some Elegant Applications of the
Pigeonhole Principle
• Ex. 10
• Ex. 11
• Suppose a1, a2, …, aN is a sequence of real
numbers, a subsequence of this sequence
is a sequence of the form ai1, ai2, …aim,
where 1<=i1<i2<…<im<=N
• Theorem 3: Every sequence of n2+1
distinct real number contains a
subsequence of length n+1 that is either
strictly increasing or strictly decreasing.
– Ex.12
– Ramsey theory
• Ex.13
• Ramsey number R(m,n): the minimum number of
people at a party such that there are either m
mutual friends or n mutual enemies
5.3 Permutations and Combinations
• Permutations
– A permutation of a set of distinct objects is an
ordered arrangement of these objects
– r-permutation: an ordered arrangement of r
elements of a set
– Ex.1
– Ex.2
– P(n,r): the number of r-permutations of a set
with n elements
• Theorem 1: If n is a positive integer and r
is an integer with 1<=r<=n, then there are
P(n,r)=n(n-1)(n-2)…(n-r+1) r-permutations
of a set with n distinct elements.
– P(n,0)=1
– P(n,n)=n!
• Corollary 1: If n and r are integers with
0<=r<=n, then P(n,r)=n!/(n-r)!
– Ex. 4-7
• Combinations
– Finding the number of subsets of a particular
size
– r-combination: an unordered selection of r
elements from the set
– Ex.8-9
– C(n,r): the number of r-combinations of a set
with n elements
• Also denoted by
coefficient
n
 
r
, and is called a binomial
• Theorem 2: The number of r-combinations of a
set with n elements, where n is a nonnegative
integer and r is an integer with 0<=r<=n, equals
C(n,r)=n!/(r!(n-r)!)
– Proof
– C(n,r)=n(n-1)…(n-r+1)/r!
• Corollary 2: Let n and r be nonnegative integers
with r<=n. Then C(n,r)=C(n,n-r).
– Proof
• Definition 1: A combinatorial proof of an
identity: using counting arguments to
prove that both sides of the identity count
the same objects but in different ways.
– Ex.12-15
5.4 Binomial Coefficients
• (x+y)n
– Ex.1
• Theorem 1: (The Binomial Theorem) Let x and y
be variables, and let n be a nonnegative integer.
Then,
(x+y)n=j=0..n C(n,r)xn-jyj
=C(n,0)xn+C(n,1)xn-1y+…+C(n,n-1)xyn1+C(n,n)yn
– Proof (combinatorial proof)
– Ex. 2-4
• Corollary 1: Let n be a nonnegative integer. Then,
k=0..nC(n,k)=2n.
– Proof
• Corollary 2: Let n be a positive integer. Then,
k=0..n (-1)kC(n,k)=0.
– Proof
– C(n,0)+C(n,2)+C(n,4)+…=C(n,1)+C(n,3)+C(n,5)+…
• Corollary 3: Let n be a nonnegative integer. Then,
k=0..n 2k C(n,k)=3n.
– Proof
Pascal’s Identity and Triangle
• Theorem 2: (Pascal’s Identity) Let n and k
be positive integers with n>=k. Then,
C(n+1,k)=C(n,k-1)+C(n,k).
– Proof
– This can be used to recursively define
binomial coefficients.
• Pascal’s triangle
– C(n,k), k=0, 1, …, n
FIGURE 1 (5.4)
FIGURE 1 Pascal’s Triangle.
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Some Other Identities of the Binomial
Coefficients
• Theorem 3: (Vandermonde’s Identity) Let m, n,
and r be nonnegative integers with r not
exceeding either m or n. Then,
C(m+n, r)=k=0..rC(m,r-k)C(n,k).
– Proof
• Corollary 4: If n is a nonnegative integer, then,
C(2n,n)=k=0..nC(n,k)2
– Proof
• Theorem 4: Let n and r be nonnegative integers
with r<=n. Then,
C(n+1,r+1)= j=r..nC(j,r).
– Proof
5.5 Generalized Permutations and
Combinations
• Permutations with repetition
– Theorem 1: The number of r-permutations of a set of
n objects with repetition allowed is nr.
– Ex.1
• Combinations with repetition
– Theorem 2: There are C(n+r-1,r)=C(n+r-1,n-1) rcombinations from a set with n elements when
repetition of elements is allowed.
– Ex.2-3
– Ex.4-6
FIGURE 1 (5.5)
FIGURE 1 Cash Box with Seven Types of Bills.
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FIGURE 2 (5.5)
FIGURE 2 Examples of Ways to Select Five Bills.
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TABLE 1 (5.5)
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• Permutations with indistinguishable
objects
– Theorem 3: The number of different
permutations of n objects, where there are n1
indistinguishable objects of type 1, n2
indistinguishable objects of type 2, …, and nk
indistinguishable objects of type k, is
n!/(n1!n2!...nk!).
– Ex.7
• Distributing objects into boxes
– Objects: distinguishable/indistinguishable
– Boxes: distinguishable/indistinguishable
• Distinguishable objects and distinguishable
boxes
– Ex.8
– Theorem 4: the number of ways to distribute n
distinguishable objects into k distinguishable boxes so
that ni objects are placed into box i, i=1, 2, …, k,
equals
n!/(n1!n2!...nk!).
• Indistinguishable objects and
distinguishable boxes
– The same as counting the number of ncombinations for a set with k elements when
repetitions are allowed
• Ex.9
• Distinguishable objects and
indistinguishable boxes
– More difficult, no simple closed formula
– Ex.10
• Indistinguishable objects and
indistinguishable boxes
– No simple closed formula
– Ex.11
– The same as partitioning positive integer n
into k positive integers.
5.6 Generating Permutations and
Combinations
• Generating permutations
– Lexicographic (dictionary) ordering
• Permutation a1a2…an precedes b1b2…bn, if for some k, with
1<=k<=n, a1=b1, a2=b2, …, ak-1=bk-1, and ak<bk.
• Ex.1
• For a1a2…ajaj+1…an such that
aj<aj+1
aj+1>aj+2>…>an
– Find the smallest among aj+1, …, an that is > aj
– Increasing order for the remaining numbers
• Ex.2
• Ex.3
• Algorithm 1: Generating the next permutation in
lexicographic order
– Procedure next_permutation(a1a2…an)
j:=n-1
while aj>aj+1
j:=j-1
k:=n
while aj>ak
k:=k-1
interchange aj and ak
r:=n
s:=j+1
while r>s
begin
interchange ar and as
r:=r-1
s:=s+1
end
• Generating Combinations
– Correspondence with bit strings of length n
– Binary expansion of an integer between 0 and
2n-1
– At each stage, the next binary expansion is
found by locating the first position from the
right that is not a 1, then changing all the 1s to
the right of this position to 0s and making this
first 0 a 1.
• Ex.4
• Algorithm 2: Generating the next larger bit
string
– Procedure next_bit_string(bn-1bn-2…b1b0)
i:=0
while bi=1
begin
bi:=0
i:=i+1
end
• The r-combinations can be listed using
lexicographic order on the sequence
– The next combinations after a1a2…ar can be
obtained by:
• Locate the last ai such that ai!=n-r+i
• Replace ai with ai+1, aj with ai+j-i+1, for j=i+1, …, r
• Ex.5
• Algorithm 3: generating the next rcombination in lexicographic order
– Procedure next_r_comb(a1a2…ar)
i:=r
while ai=n-r+i
i:=i-1
ai:=ai+1
for j:=i+1 to r
aj:=ai+j-i
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