Chapter 6 Counting

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Chapter 6 Counting
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6.2 The Multiplication Rule
Possibility Trees If an operation consists of
several steps (such as choosing a meal), then we
can use a tree to show all the possibilities that
this operation can be performed.
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6.2 The Multiplication Rule

• Example
• If a diner offers a simple meal that consists of
• Either soup or salad
• 3 choices of entrée
• 3 choices of beverage. How many
  meals are possible?

step 1
step 2
step 3
soup
Start
salad
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6.2 The Multiplication Rule
Theorem If an operation can be broken into k
steps and the first step can be performed in n1
ways, the second step can be performed in n2
ways (regardless of the

outcome of the 1st step), .
. . the kth
step can be performed in nk ways (regardless of
the
outcomes of the previous
steps), then the entire operation can be
performed in n1 n2 nk ways.
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6.2 The Multiplication Rule
Definition A permutation of a set of objects is
an ordering of the objects in a row.
Theorem 6.2.2 For any integer n with n 1, the
number of permutations of a set of n objects is n!
Definition An r-permutation of a set of n
elements is an ordered selection of r elements
taken from the set of n elements. The number of
such r-permutations is denoted by P(n,r)
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6.2 The Multiplication Rule
Theorem If n and r are integers such that 1
r n , then
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A Simple but Powerful Counting Technique
Theorem Given two finite sets A and B. If there
is a one-to-one correspondence between them, then
n(A) n(B). In other words, if set A is
inconvenient to count, and we know that there is
a set B which is in 1-to-1 correspondence with A,
then we can count the elements in B
instead. Example Suppose that we want to find
out the number of people in a given concert, and
we know that each person has to hand over one
ticket (and no one will hand over two or more
tickets) at the entrance, then we can count
tickets collected instead of counting heads.
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6.3 The Addition Rule
Theorem If a finite set A is the union of k
mutually disjoint subsets A1, A2, , Ak, then
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Example The most common type of combination lock
is the 1500 padlock (pictured below). It has 40
numbers on it (from 0 to 39) and it can be
unlocked only if the correct 3-number code is
dialed in.
Due to its mechanism, the middle number in the
code must be bigger than the other two numbers.
In this case, what is the maximum number of such
codes are possible?
The actual number is 1500 according to the
manufacturer, much smaller than our calculations,
this may due to other restrictions.
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Theorem (The difference rule) If A is a finite
set and B is a subset of A, then
Example If you want to know the number of people
absent, we can count the number of people
present, and then subtract that number from the
total.
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6.3 The Addition Rule
Inclusion/Exclusion Rule If A, B, and C are any
three finite sets, then
and
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A
B
C
n(B)
n(C)
n(A)
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A
B
C
n(B)
n(C)
- n(AnB)
- n(AnC)
- n(BnC)
n(A)
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Example
How many integers from 1 through 1000 are either
multiples of 3 or multiples of 5?
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6.4 Combinations
Definition Let n and r be non-negative integers
with r n. An r-combination of a set of n
elements is a subset of r elements from that
set. The following symbols, all read n choose r
, are all used the denote the number of size r
subsets that can be chosen from a set of n
elements.
Theorem Let n and r be non-negative integers
with r n, then
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6.4 Combinations
Permutations of a set with Repeated elements
Example How many ways are there to re-order the
letters in the word
MISSISSIPPI ?
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6.4 Combinations
Permutations of a set with Repeated elements
Theorem Suppose a collection consists of n
objects of which n1 are of type 1 and are
indistinguishable from each other, n2 are of
type 2 and are indistinguishable from each
other, . . . nk are of type k and are
indistinguishable from each other,
and suppose that n1 n2 nk n, then the
number of permutations of these n objects can be
computed by the following formula.
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6.4 Combinations
Permutations of a set with Repeated elements
Challenging Exercise How many ways are there to
re-order the letters in the word
MISSISSIPPI such that no two Is
are next to each other?
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6.5 r-Combinations with Repetition Allowed
Example Suppose that there are 4 different types
of candies, each type is priced at 1 a piece. If
you want to spend 10 on these candies, how many
different ways can you do it? (You dont have to
include every type)
Another equivalent example Suppose that there
are 4 different containers, and 10 identical
balls. If the containers are all big enough to
hold 10 balls, how many ways can you put these 10
balls into some or all of these containers?
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6.5 r-Combinations with Repetition Allowed
Definition An r-combination with repetition
allowed, chosen from a set X of n elements is an
unordered selection of elements taken from X with
repetition allowed. (note that r can be n.) If
X x1, x2 , , xn , we write an
r-combination with repetition allowed as y1, y2
, , yr where each yi is equal to a xj for
some j.
Theorem The number of r-combinations with
repetition allowed , that can be selected from a
set of n objects is