CHAPTER 6 (TAN) Set Counting
CHAPTER 6 (TAN)
Set & Counting
MA1103
Business Mathematics I
6.1 SETS AND SET OPERATIONS
2
Set
A set is a well-defined collection of objects.
The objects of a set are called the elements, or members, of a
set and are usually denoted by lowercase letters a, b, c, . . . ; the
sets themselves are usually denoted by uppercase letters A, B, C,
...
If a is an element of a set A, we write a A and read “a belongs
to A” or “a is an element of A.”
If the element a does not belong to the set A, however, then we
write a A and read “a does not belong to A.”
For example, if A = {1, 2, 3, 4, 5}, then 3 A but 6 A.
3
Set Notations
Roster notation
• The set A consisting of the first three letters of the
English alphabet is written A = {a, b, c}
• The set B of all letters of the alphabet can be written
B = {a, b, c, . . . , z}
Set-builder notation
B = {x | x is a letter of the English alphabet}, is read “B
is the set of all elements x such that x is a letter of the
English alphabet.”
4
Set Equality
Two sets A and B are equal, written A = B, if and only if they have
exactly the same elements.
If every element of a set A is also an element of a set B, then we
say that A is a subset of B and write A B.
Example.
A = {a, e, i, o, u}, B = {a, i, o, e, u}, C = {a, e, i, o}
Which sets are equal? Which set is a subset of another set?
The set that contains no elements is called the empty set and is
denoted by .
Example. List all subsets of the set A {a, b, c}.
5
Venn Diagrams
A universal set is the set of all elements of
interest in a particular discussion. It is the
largest in the sense that all sets considered in
the discussion of the problem are subsets of the
universal set.
A visual representation of sets is realized
through the use of Venn diagrams.
6
Venn Diagram of
Equality and Subset
7
Set Operations
8
Properties of Set Operations
9
An Example
Let U denote the set of all cars in a dealer’s lot, and let
A = {x U 0 x is equipped with Sirius XM Radio}, B = {x U
0 x is equipped with a moonroof}, C = {x U 0 x is
equipped with side air bags}.
Find an expression in terms of A, B, and C for each of
the following sets:
a. The set of cars with at least one of the given options
b. The set of cars with exactly one of the given options
c. The set of cars with Sirius XM Radio and side air bags
but no moonroof.
10
6.2 THE NUMBER OF ELEMENTS IN
SET
11
Counting & Combinatorics
The solution to some problems in mathematics calls for
finding the number of elements in a set.
Such problems are called counting problems and
constitute a field of study known as combinatorics.
The number of elements in a finite set is determined by
simply counting the elements in the set.
If A is a set, then n(A) denotes the number of elements
in A.
12
Union of Sets
Example.
If A = {a, c, d} and B = {b, e, f, t}, then n(A) = 3 and n(B) = 4, so n(A) + n(B) = 7.
Moreover, A B = {a, b, c, d, e, f, t} and n(A B) = 7.
If A and B are disjoint sets, then
n(A B) = n(A) + n(B)
In general case,
n(A B) = n(A) + n(B) - n(A B)
Example.
In a survey of 100 coffee drinkers, it was found that 70 take sugar, 60 take
cream, and 50 take both sugar and cream with their coffee. How many coffee
drinkers take sugar or cream with their coffee?
13
Another Example
A leading cosmetics manufacturer advertises its products in three magazines: Allure,
Cosmopolitan, and the Ladies Home Journal. A survey of 500 customers by the
manufacturer reveals the following information:
180 learned of its products from Allure.
200 learned of its products from Cosmopolitan.
192 learned of its products from the Ladies Home Journal.
84 learned of its products from Allure and Cosmopolitan.
52 learned of its products from Allure and the Ladies Home Journal.
64 learned of its products from Cosmopolitan and the Ladies Home Journal.
38 learned of its products from all three magazines.
How many of the customers saw the manufacturer’s advertisement in
a. At least one magazine?
b. Exactly one magazine?
14
6.3 THE MULTIPLICATION PRINCIPLE
15
The Multiplication Principle
Suppose there are m ways of performing a task T1 and n ways of
performing a task T2. Then there are mn ways of performing the
task T1 followed by the task T2.
Example.
Three trunk roads connect Town A and Town B, and two trunk
roads connect Town B and Town C.
a. Use the multiplication principle to find the number of ways in
which a journey from Town A to Town C via Town B can be
completed.
b. Verify part (a) directly by exhibiting all possible routes.
16
More Examples
1. Menu Choices Diners at Angelo’s Spaghetti Bar can
select their entree from 6 varieties of pasta and 28
choices of sauce. How many such combinations are
there that consist of 1 variety of pasta and 1 kind of
sauce?
2. Chairs in an auditorium are labeled with one capital
letter followed by a positive integer at most 100.
How many chairs can be labeled differently?
17
Generalized Multiplication Principle
Example.
A coin is tossed three times, and the sequence of heads and tails is recorded.
a. Use the generalized multiplication principle to determine the number of
possible outcomes of this activity.
b. Exhibit all the sequences by means of a tree diagram.
18
More Examples
1. How many different car plates could be made by using exactly one letter, three
decimal digit, and then two other letters?
2. A combination lock is unlocked by dialing a sequence of numbers: first to the left,
then to the right, and to the left again. If there are ten digits on the dial,
determine the number of possible combinations.
3. An investor has decided to purchase shares in the stock of three companies: one
engaged in aerospace activities, one involved in energy development, and one
involved in electronics. After some research, the account executive of a
brokerage firm has recommended that the investor consider stock from five
aerospace companies, three energy development companies, and four
electronics companies. In how many ways can the investor select the group of
three companies from the executive’s list?
4. Tom is planning to leave for New York City from Washington, D.C., on Monday
morning and has decided that he will either fly or take the train. There are five
flights and two trains departing for New York City from Washington that morning.
When he returns on Sunday afternoon, Tom plans to either fly or hitch a ride
with a friend. There are two flights departing from New York City to Washington
that afternoon. In how many ways can Tom complete this round trip?
19
6.4 PERMUTATIONS &
COMBINATIONS
20
Permutation
A permutation of the set is an arrangement of these objects in a
definite order.
The order in which objects are arranged is important!
Example.
Let A {a, b, c}.
a. Find the number of permutations of A.
b. List all the permutations of A with the aid of a tree diagram.
21
An Example
Find the number of ways in which a baseball team consisting of nine
people can arrange themselves in a line for a group picture.
We can derive an expression for the number of ways of permuting a
set A
of n distinct objects taken n at a time. In fact, each permutation may
be viewed as being obtained by filling each of n blanks with one and
only one element from the set. There are n ways of filling the first
blank, followed by (n-1) ways of filling the second blank, and so on.
Thus, by the generalized multiplication principle, there are
n . n - 1 . n- 2 . . . 3 . 2 . 1
ways of permuting the elements of the set A.
22
Factorial and Permutation
The number of permutations of n distinct objects taken n at a time, denoted
by P(n, n), is
P(n,n) = n!
The number of ways of permuting n distinct objects taken r at a time,
denoted by P(n, r), is given by
23
Examples
1. Compute
(a) P(4, 4) and
(b) P(4, 2), and interpret your results.
2. Find the number of ways in which a
chairman, a vice-chairman, a secretary, and a
treasurer can be chosen from a committee of
eight members.
24
Permutations of n Objects,
Not All Distinct
Given a set of n objects in which n1 objects are alike and of one
kind, n2 objects are alike and of another kind, . . . , and nm
objects are alike and of yet another kind, so that n1 + n2 + … + nm
= n.
To count the number of permutations of these n objects taken n
at a time, denote the number of such permutations by x.
If we think of the n1 objects as being distinct, then they can be
permuted in n1! ways. Similarly, if we think of the n2 objects as
being distinct, then they can be permuted in n2! ways, and so on.
Therefore, if we think of the n objects as being distinct, then, by
the generalized multiplication principle, there are x . n1! . n2! …
nm! permutations of these objects.
But, x . n1! . n2! … nm! = n!
25
Examples
1. Find the number of permutations that can be formed from all
the letters in the word ATLANTA.
2. Management Decisions Weaver and Kline, a stock brokerage
firm, has received nine inquiries regarding new accounts. In
how many ways can these inquiries be directed to any three
of the firm’s account executives if each account executive is
to handle three inquiries?
26
Combination
Combination of a set is a an arrangement of r objects
from a set of n objects without any regard to the order
in which the objects are selected.
27
Counting Combination
𝐶 𝑛, 𝑟 or 𝑛𝑟 is the number of combinations of n objects
taken r at a time.
Each of the C(n, r) combinations of r objects can be permuted
in r! ways.
The product r! C(n, r) gives the number of permutations of n
objects taken r at a time; that is, r! C(n, r) = P(n, r).
28
Examples
1. Compute and interpret the results of
(a) C(4, 4) and (b) C(4, 2).
2. A Senate investigation subcommittee of four
members is to be selected from a Senate
committee of ten members. Determine the
number of ways in which this can be done.
3. How many poker hands of 5 cards can be
dealt from a standard deck of 52 cards?
29
More Than One Way of Counting
1. The members of a string quartet consisting of two violinists, a
violist, and a cellist are to be selected from a group of six violinists,
three violists, and two cellists.
a. In how many ways can the string quartet be formed?
b. In how many ways can the string quartet be formed if one of the violinists is to
be designated as the first violinist and the other is to be designated as the second
violinist?
2. The Futurists, a rock group, are planning a concert tour with
performances to be given in five cities: San Francisco, Los Angeles,
San Diego, Denver, and Las Vegas. In how many ways can they
arrange their itinerary if
a. There are no restrictions?
b. The three performances in California must be given consecutively?
30
More Than One Way of Counting
3. The United Nations Security Council consists
of 5 permanent members and 10
nonpermanent members. Decisions made by
the council require 9 votes for passage.
However, any permanent member may veto
a measure and thus block its passage.
Assuming that there are no abstentions, in
how many ways can a measure be passed if
all 15 members of the Council vote?
31
Set & Counting
MA1103
Business Mathematics I
6.1 SETS AND SET OPERATIONS
2
Set
A set is a well-defined collection of objects.
The objects of a set are called the elements, or members, of a
set and are usually denoted by lowercase letters a, b, c, . . . ; the
sets themselves are usually denoted by uppercase letters A, B, C,
...
If a is an element of a set A, we write a A and read “a belongs
to A” or “a is an element of A.”
If the element a does not belong to the set A, however, then we
write a A and read “a does not belong to A.”
For example, if A = {1, 2, 3, 4, 5}, then 3 A but 6 A.
3
Set Notations
Roster notation
• The set A consisting of the first three letters of the
English alphabet is written A = {a, b, c}
• The set B of all letters of the alphabet can be written
B = {a, b, c, . . . , z}
Set-builder notation
B = {x | x is a letter of the English alphabet}, is read “B
is the set of all elements x such that x is a letter of the
English alphabet.”
4
Set Equality
Two sets A and B are equal, written A = B, if and only if they have
exactly the same elements.
If every element of a set A is also an element of a set B, then we
say that A is a subset of B and write A B.
Example.
A = {a, e, i, o, u}, B = {a, i, o, e, u}, C = {a, e, i, o}
Which sets are equal? Which set is a subset of another set?
The set that contains no elements is called the empty set and is
denoted by .
Example. List all subsets of the set A {a, b, c}.
5
Venn Diagrams
A universal set is the set of all elements of
interest in a particular discussion. It is the
largest in the sense that all sets considered in
the discussion of the problem are subsets of the
universal set.
A visual representation of sets is realized
through the use of Venn diagrams.
6
Venn Diagram of
Equality and Subset
7
Set Operations
8
Properties of Set Operations
9
An Example
Let U denote the set of all cars in a dealer’s lot, and let
A = {x U 0 x is equipped with Sirius XM Radio}, B = {x U
0 x is equipped with a moonroof}, C = {x U 0 x is
equipped with side air bags}.
Find an expression in terms of A, B, and C for each of
the following sets:
a. The set of cars with at least one of the given options
b. The set of cars with exactly one of the given options
c. The set of cars with Sirius XM Radio and side air bags
but no moonroof.
10
6.2 THE NUMBER OF ELEMENTS IN
SET
11
Counting & Combinatorics
The solution to some problems in mathematics calls for
finding the number of elements in a set.
Such problems are called counting problems and
constitute a field of study known as combinatorics.
The number of elements in a finite set is determined by
simply counting the elements in the set.
If A is a set, then n(A) denotes the number of elements
in A.
12
Union of Sets
Example.
If A = {a, c, d} and B = {b, e, f, t}, then n(A) = 3 and n(B) = 4, so n(A) + n(B) = 7.
Moreover, A B = {a, b, c, d, e, f, t} and n(A B) = 7.
If A and B are disjoint sets, then
n(A B) = n(A) + n(B)
In general case,
n(A B) = n(A) + n(B) - n(A B)
Example.
In a survey of 100 coffee drinkers, it was found that 70 take sugar, 60 take
cream, and 50 take both sugar and cream with their coffee. How many coffee
drinkers take sugar or cream with their coffee?
13
Another Example
A leading cosmetics manufacturer advertises its products in three magazines: Allure,
Cosmopolitan, and the Ladies Home Journal. A survey of 500 customers by the
manufacturer reveals the following information:
180 learned of its products from Allure.
200 learned of its products from Cosmopolitan.
192 learned of its products from the Ladies Home Journal.
84 learned of its products from Allure and Cosmopolitan.
52 learned of its products from Allure and the Ladies Home Journal.
64 learned of its products from Cosmopolitan and the Ladies Home Journal.
38 learned of its products from all three magazines.
How many of the customers saw the manufacturer’s advertisement in
a. At least one magazine?
b. Exactly one magazine?
14
6.3 THE MULTIPLICATION PRINCIPLE
15
The Multiplication Principle
Suppose there are m ways of performing a task T1 and n ways of
performing a task T2. Then there are mn ways of performing the
task T1 followed by the task T2.
Example.
Three trunk roads connect Town A and Town B, and two trunk
roads connect Town B and Town C.
a. Use the multiplication principle to find the number of ways in
which a journey from Town A to Town C via Town B can be
completed.
b. Verify part (a) directly by exhibiting all possible routes.
16
More Examples
1. Menu Choices Diners at Angelo’s Spaghetti Bar can
select their entree from 6 varieties of pasta and 28
choices of sauce. How many such combinations are
there that consist of 1 variety of pasta and 1 kind of
sauce?
2. Chairs in an auditorium are labeled with one capital
letter followed by a positive integer at most 100.
How many chairs can be labeled differently?
17
Generalized Multiplication Principle
Example.
A coin is tossed three times, and the sequence of heads and tails is recorded.
a. Use the generalized multiplication principle to determine the number of
possible outcomes of this activity.
b. Exhibit all the sequences by means of a tree diagram.
18
More Examples
1. How many different car plates could be made by using exactly one letter, three
decimal digit, and then two other letters?
2. A combination lock is unlocked by dialing a sequence of numbers: first to the left,
then to the right, and to the left again. If there are ten digits on the dial,
determine the number of possible combinations.
3. An investor has decided to purchase shares in the stock of three companies: one
engaged in aerospace activities, one involved in energy development, and one
involved in electronics. After some research, the account executive of a
brokerage firm has recommended that the investor consider stock from five
aerospace companies, three energy development companies, and four
electronics companies. In how many ways can the investor select the group of
three companies from the executive’s list?
4. Tom is planning to leave for New York City from Washington, D.C., on Monday
morning and has decided that he will either fly or take the train. There are five
flights and two trains departing for New York City from Washington that morning.
When he returns on Sunday afternoon, Tom plans to either fly or hitch a ride
with a friend. There are two flights departing from New York City to Washington
that afternoon. In how many ways can Tom complete this round trip?
19
6.4 PERMUTATIONS &
COMBINATIONS
20
Permutation
A permutation of the set is an arrangement of these objects in a
definite order.
The order in which objects are arranged is important!
Example.
Let A {a, b, c}.
a. Find the number of permutations of A.
b. List all the permutations of A with the aid of a tree diagram.
21
An Example
Find the number of ways in which a baseball team consisting of nine
people can arrange themselves in a line for a group picture.
We can derive an expression for the number of ways of permuting a
set A
of n distinct objects taken n at a time. In fact, each permutation may
be viewed as being obtained by filling each of n blanks with one and
only one element from the set. There are n ways of filling the first
blank, followed by (n-1) ways of filling the second blank, and so on.
Thus, by the generalized multiplication principle, there are
n . n - 1 . n- 2 . . . 3 . 2 . 1
ways of permuting the elements of the set A.
22
Factorial and Permutation
The number of permutations of n distinct objects taken n at a time, denoted
by P(n, n), is
P(n,n) = n!
The number of ways of permuting n distinct objects taken r at a time,
denoted by P(n, r), is given by
23
Examples
1. Compute
(a) P(4, 4) and
(b) P(4, 2), and interpret your results.
2. Find the number of ways in which a
chairman, a vice-chairman, a secretary, and a
treasurer can be chosen from a committee of
eight members.
24
Permutations of n Objects,
Not All Distinct
Given a set of n objects in which n1 objects are alike and of one
kind, n2 objects are alike and of another kind, . . . , and nm
objects are alike and of yet another kind, so that n1 + n2 + … + nm
= n.
To count the number of permutations of these n objects taken n
at a time, denote the number of such permutations by x.
If we think of the n1 objects as being distinct, then they can be
permuted in n1! ways. Similarly, if we think of the n2 objects as
being distinct, then they can be permuted in n2! ways, and so on.
Therefore, if we think of the n objects as being distinct, then, by
the generalized multiplication principle, there are x . n1! . n2! …
nm! permutations of these objects.
But, x . n1! . n2! … nm! = n!
25
Examples
1. Find the number of permutations that can be formed from all
the letters in the word ATLANTA.
2. Management Decisions Weaver and Kline, a stock brokerage
firm, has received nine inquiries regarding new accounts. In
how many ways can these inquiries be directed to any three
of the firm’s account executives if each account executive is
to handle three inquiries?
26
Combination
Combination of a set is a an arrangement of r objects
from a set of n objects without any regard to the order
in which the objects are selected.
27
Counting Combination
𝐶 𝑛, 𝑟 or 𝑛𝑟 is the number of combinations of n objects
taken r at a time.
Each of the C(n, r) combinations of r objects can be permuted
in r! ways.
The product r! C(n, r) gives the number of permutations of n
objects taken r at a time; that is, r! C(n, r) = P(n, r).
28
Examples
1. Compute and interpret the results of
(a) C(4, 4) and (b) C(4, 2).
2. A Senate investigation subcommittee of four
members is to be selected from a Senate
committee of ten members. Determine the
number of ways in which this can be done.
3. How many poker hands of 5 cards can be
dealt from a standard deck of 52 cards?
29
More Than One Way of Counting
1. The members of a string quartet consisting of two violinists, a
violist, and a cellist are to be selected from a group of six violinists,
three violists, and two cellists.
a. In how many ways can the string quartet be formed?
b. In how many ways can the string quartet be formed if one of the violinists is to
be designated as the first violinist and the other is to be designated as the second
violinist?
2. The Futurists, a rock group, are planning a concert tour with
performances to be given in five cities: San Francisco, Los Angeles,
San Diego, Denver, and Las Vegas. In how many ways can they
arrange their itinerary if
a. There are no restrictions?
b. The three performances in California must be given consecutively?
30
More Than One Way of Counting
3. The United Nations Security Council consists
of 5 permanent members and 10
nonpermanent members. Decisions made by
the council require 9 votes for passage.
However, any permanent member may veto
a measure and thus block its passage.
Assuming that there are no abstentions, in
how many ways can a measure be passed if
all 15 members of the Council vote?
31