CHAPTER 6 (TAN)
Set & Counting
MA1103
Set
A set is a well-defined collection of objects.
Set Equality
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
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.
An Example
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
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.
Union of Sets
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?
The Multiplication Principle
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.
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.
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.
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?
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}.
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
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!
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
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
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
Combination
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
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.
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?
