APPENDIX D
Counting Principles and Probability
D1
Appendix D Counting Principles and Probability
D.1
Counting Principles
Simple Counting Problems • Counting Principles •
Permutations • Combinations
Simple Counting Problems
You have already learned some of the basic counting principles and their application to probability. In the next section, you will see that much of probability has
to do with counting the number of ways an event can occur. Examples 1, 2, and
3 describe some simple cases.
EXAMPLE 1 A Random Number Generator
A random number generator (on a computer) selects an integer from 1 to 30. Find
the number of ways each event can occur.
(a) An even integer is selected.
(b) An integer that is less than 12 is selected.
(c) A prime number is selected.
(d) A perfect square is selected.
Solution
(a) Because half of the integers from 1 to 30 are even, this event can occur in 15
different ways.
(b) The integers from 1 to 30 that are less than 12 are as follows.
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11
Because this set has 11 members, you can conclude that there are 11 different ways this event can occur.
(c) The prime numbers from 1 to 30 are as follows.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Because this set has 10 members, you can conclude that there are 10 different ways this event can occur.
(d) The perfect squares from 1 to 30 are as follows.
1, 4, 9, 16, 25
Because this set has 5 members, you can conclude that there are 5 different
ways this event can occur.
NOTE A program for generating random numbers for several models of calculators can be found in Appendix H.
D2
APPENDIX D
Counting Principles and Probability
EXAMPLE 2
Selecting Pairs of Numbers at Random
Eight pieces of paper are numbered from 1 to 8 and placed in a box. One piece of
paper is drawn from the box, its number is written down, and the piece of paper
is replaced in the box. Then, a second piece of paper is drawn from the box, and
its number is written down. Finally, the two numbers are added together. How
many different ways can a total of 12 be obtained?
Solution
To solve this problem, count the different ways that a total of 12 can be obtained
using two numbers from 1 to 8.
First number Second number 12
After considering the various possibilities, you can see that this equation can be
solved in the following five ways.
First Number:
4 5 6 7 8
Second Number: 8 7 6 5 4
So, a total of 12 can be obtained in five different ways.
Solving counting problems can be tricky. Often, seemingly minor changes in
the statement of a problem can affect the answer. For instance, compare the
counting problem in the next example with that given in Example 2.
EXAMPLE 3 Selecting Pairs of Numbers at Random
NOTE The difference between
the counting problems in
Examples 2 and 3 can be
distinguished by saying that the
random selection in Example 2
occurs with replacement, whereas the random selection in
Example 3 occurs without
replacement, which eliminates
the possibility of choosing two
6’s.
Eight pieces of paper are numbered from 1 to 8 and placed in a box. Two pieces
of paper are drawn from the box, and the numbers on them are written down and
totaled. How many different ways can a total of 12 be obtained?
Solution
To solve this problem, count the different ways that a total of 12 can be obtained
using two different numbers from 1 to 8.
First number Second number 12
After considering the various possibilities, you can see that this equation can be
solved in the following four ways.
First Number:
4 5 7 8
Second Number: 8 7 5 4
So, a total of 12 can be obtained in four different ways.
APPENDIX D
Counting Principles and Probability
D3
Counting Principles
The first three examples in this section are considered simple counting problems
in which you can list each possible way that an event can occur. When it is possible, this is always the best way to solve a counting problem. However, some
events can occur in so many different ways that it is not feasible to write out the
entire list. In such cases, you must rely on formulas and counting principles. The
most important of these is called the Fundamental Counting Principle.
NOTE The Fundamental
Counting Principle can be extended to three or more events. For
instance, the number of ways that
three events E1, E2, and E3 can
occur is m1 m2 m3.
Additional problem: After completing Example 4, ask students to
determine how many different pairs
of letters from the English alphabet
are possible if the two letters must
be different.
Fundamental Counting Principle
Let E1 and E2 be two events. The first event E1 can occur in m1 different
ways. After E1 has occurred, E2 can occur in m2 different ways. The
number of ways that the two events can occur is
m1
m2.
EXAMPLE 4 Applying the Fundamental Counting Principle
The English alphabet contains 26 letters. So, the number of possible “two-letter
words” is 26 26 676.
Answer: 650
EXAMPLE 5
Applying the Fundamental Counting Principle
Telephone numbers in the United States have 10 digits. The first three are the area
code and the next seven are the local telephone number. How many different telephone numbers are possible within each area code? (A telephone number cannot
have 0 or 1 as its first or second digit.)
Solution
There are only eight choices for the first and second digits because neither can be
0 or 1. For each of the other digits, there are 10 choices.
Local telephone number
Area code
)
)
−
8
8
10
10
10
10
10
So, by the Fundamental Counting Principle, the number of local telephone numbers that are possible within each area code is
8
8 10 10 10 10 10 6,400,000.
D4
APPENDIX D
Counting Principles and Probability
Permutations
One important application of the Fundamental Counting Principle is in determining the number of ways that n elements can be arranged (in order). An ordering
of n elements is called a permutation of the elements.
Definition of Permutation
A permutation of n different elements is an ordering of the elements
such that one element is first, one is second, one is third, and so on.
EXAMPLE 6 Listing Permutations
The six possible permutations of letters A, B, and C are as follows.
A, B, C
B, A, C
C, A, B
A, C, B
B, C, A
C, B, A
EXAMPLE 7 Finding the Number of Permutations of n Elements
How many permutations are possible for the letters A, B, C, D, E, and F?
Solution
First position:
Second position:
Third position:
Fourth position:
Fifth position:
Sixth position:
Any of the six letters.
Any of the remaining five letters.
Any of the remaining four letters.
Any of the remaining three letters.
Any of the remaining two letters.
The one remaining letter.
So, the numbers of choices for the six positions are as follows.
Permutations of six letters
6
5
4
3
2
1
By the Fundamental Counting Principle, the total number of permutations of the
six letters is
6
5 4 3 2 1 6! 720.
APPENDIX D
Counting Principles and Probability
D5
The result obtained in Example 7 is generalized below.
Number of Permutations of n Elements
The number of permutations of n elements is given by
n
n 1 .
. .
4 3 2 1 n!.
In other words, there are n! different ways that n elements can be ordered.
EXAMPLE 8 Finding the Number of Permutations of n Elements
How many ways can you form a four-digit number using each of the digits 1, 3,
5, and 7 exactly once?
Solution
One way to solve this problem is simply to list the number of ways.
1357, 1375, 1537, 1573, 1735, 1753
3157, 3175, 3517, 3571, 3715, 3751
5137, 5173, 5317, 5371, 5713, 5731
7135, 7153, 7315, 7351, 7513, 7531
Another way to solve the problem is to use the formula for the number of
permutations of four elements. By that formula, there are 4! 24 permutations.
EXAMPLE 9 Finding the Number of Permutations
of n Elements
You are a supervisor for 11 different employees. One of your responsibilities is to
perform an annual evaluation for each employee, and then rank the performances
of the 11 different employees. How many different rankings are possible?
Solution
You have 11 choices for the first ranking. After choosing the first ranking, you
can choose any of the remaining 10 for the second ranking, and so on.
Rankings of 11 employees
11
10
9
8
7
6
5
4
3
2
1
So, the number of different rankings is 11! 39,916,800.
D6
APPENDIX D
Counting Principles and Probability
Combinations
When counting the number of possible permutations of a set of elements, order is
important. The final topic in this section is a method of selecting subsets of a larger set in which order is not important. Such subsets are called combinations of n
elements taken r at a time. For instance, the combinations
Students often confuse a permutation with a combination. Help them
see the difference using comparison
and contrast. A, B, C and B, A, C
are distinct as a permutation, but
identical as a combination.
A, B, C
and
B, A, C
are equivalent because both sets contain the same three elements, and the order in
which the elements are listed is not important. So, you would count only one of
the two sets. A common example of how a combination occurs is a card game in
which the player is free to reorder the cards after they have been dealt.
EXAMPLE 10
Combinations of n Elements Taken r at a Time
In how many different ways can three letters be chosen from the letters A, B, C,
D, and E? (The order of the three letters is not important.)
Solution
The following subsets represent the different combinations of three letters that
can be chosen from five letters.
A, B, C
A, B, E
A, C, E
B, C, D
B, D, E
A, B, D
A, C, D
A, D, E
B, C, E
C, D, E
From this list, you can conclude that there are 10 different ways that three letters
can be chosen from five letters. Because order is not important, sets such as
B, C, A are not chosen, because the set B, C, A is represented by the set
A, B, C.
The formula for the number of combinations of n elements taken r at a time
is as follows.
Number of Combinations of n Elements
Taken r at a Time
The number of combinations of n elements taken r at a time is
nCr
n!
.
n r!r!
APPENDIX D
Counting Principles and Probability
D7
Note that the formula for nCr is the same one given for binomial coefficients.
To see how this formula is used, consider the counting problem in Example 10.
In that problem, you need to find the number of combinations of five elements
taken three at a time. So, n 5, r 3, and the number of combinations is
5C3
5!
2!3!
54
32
10
3
1
which is the same as the answer obtained in the example.
EXAMPLE 11 Combinations of n Elements Taken r
at a Time
A standard poker hand consists of five cards dealt from a deck of 52. How many
different poker hands are possible? (After the cards are dealt, the player may
reorder them, and therefore order is not important.)
Solution
Use the formula for the number of combinations of 52 elements taken five at a
time, as follows.
52C5
52!
47!5!
52 51 50 49 48
54321
2,598,960
So, there are almost 2.6 million different hands. Use a graphing utility to verify
this result.
When solving problems involving counting principles, you need to be able to
distinguish among the various counting principles in order to determine which is
necessary to solve the problem correctly. To do this, ask yourself the following
questions.
1. Is the order of the elements important? Permutation
2. Are the chosen elements a subset of a larger set in which order is not important? Combination
3. Does the problem involve two or more separate events?
Counting Principle
Fundamental
D8
APPENDIX D
Counting Principles and Probability
Exercises
D.1
Random Selection In Exercises 1–6, find the number of ways the specified event can occur when one
or more marbles are selected from a bowl containing ten marbles numbered 0 through 9.
0
1
2
3
4
5
6
7
8
9
1. One marble is drawn and its number is even.
2. One marble is drawn and its number is prime.
3. Two marbles are drawn one after the other. The first
is replaced before the second is drawn. The sum of
the numbers is 10.
4. Two marbles are drawn one after the other. The first
is replaced before the second is drawn. The sum of
the numbers is 7.
5. Two marbles are drawn without replacement. The
sum of the numbers is 10.
6. Two marbles are drawn without replacement. The
sum of the numbers is 7.
In Exercises 7–16, find the
number of ways the specified event can occur when
one or more marbles are selected from a bowl
containing 20 marbles numbered 1 through 20.
Random Selection
7.
8.
9.
10.
One marble is drawn and its number is odd.
One marble is drawn and its number is even.
One marble is drawn and its number is prime.
One marble is drawn and its number is greater than
12.
11. One marble is drawn and its number is divisible by
3.
12. One marble is drawn and its number is divisible by
6.
13. Two marbles are drawn one after the other. The first
is replaced before the second is drawn. The sum of
the numbers is 8.
14. Two marbles are drawn one after the other. The first
is replaced before the second is drawn. The sum of
the numbers is 15.
15. Two marbles are drawn without replacement. The
sum of the numbers is 8.
16. Three marbles are drawn one after another. Each
marble is replaced before the next is drawn. The sum
of the numbers is 15.
17. Staffing Choices A small grocery store needs to
open another checkout line. Three people who can
run the cash register are available and two people are
available to bag groceries. How many different ways
can the additional checkout line be staffed?
18. Computer System You are in the process of
purchasing a new computer system. You must choose
one of the three monitors, one of two computers, and
one of two keyboards. How many different configurations of the system are available to you?
19. Identification Numbers In a statistical study, each
participant was given an identification label consisting of a letter of the alphabet followed by a single
digit (0 is a digit). How many distinct identification
labels can be made in this way?
20. Identification Numbers How many identification
labels (see Exercise 19) can be made consisting of a
letter of the alphabet followed by a two-digit
number?
21. License Plates How many distinct automobile
license plates can be formed using a four-digit
number followed by two letters?
22. Three-Digit Numbers How many three-digit
numbers can be formed in each of the following
situations?
(a) The hundreds digit cannot be 0.
(b) No repetition of digits is allowed.
(c) The number cannot be greater than 400.
APPENDIX D
23. Toboggan Ride Five people line up on a toboggan
at the top of a hill. In how many ways can they be
seated if only two of the five are willing to sit in the
front seat?
24. Task Assignment Four people are assigned to four
different tasks. In how many ways can the assignments be made if one of the four is not qualified for
the first task?
25. Taking a Trip Five people are taking a long trip in
a car. Two sit in the front seat and three sit in the
back seat. Three of the people agree to share the
driving. In how many different ways can the five
people sit?
26. Aircraft Boarding Eight people are boarding an
aircraft. Three have tickets for first class and board
before those in the economy class. In how many
different ways can the eight people board the
aircraft?
27. Permutations List all the permutations of the letters
X, Y, and Z.
28. Permutations List all the permutations of the letters
A, B, C, and D.
29. Seating Arrangement In how many ways can five
children be seated in a single row of chairs?
30. Seating Arrangement In how many ways can six
people be seated in a six-passenger car?
31. Choosing Officers From a pool of 10 candidates,
the offices of president, vice-president, secretary,
and treasurer will be filled. In how many ways can
the offices be filled if each of the 10 candidates can
hold any one of the offices?
32. Time Management Study There are eight steps in
accomplishing a certain task and these steps can be
performed in any order. Management wants to test
each possible order to determine which is the least
time-consuming.
(a) How many different orders will have to be tested?
(b) How many different orders will have to be tested
if one step in accomplishing the task must be done
first? (The other seven steps can be performed in
any order.)
Counting Principles and Probability
D9
33. Combination Lock A combination lock will open
when the right choice of three numbers (from 1 to
40, inclusive) is selected in order. How many
different lock combinations are possible?
34. Work Assignments Out of eight workers five are
selected and assigned to different tasks. In how many
ways can this be done assuming there are no restrictions in making the assignments?
35. Number of Subsets List all the subsets with two
elements that can be formed from the set of letters
A, B, C, D, E, F.
36. Number of Subsets List all the subsets with three
elements that can be formed from the set of letters
A, B, C, D, E, F.
37. Committee Selection Three students are selected
from a class of 20 to form a fund-raising committee.
Use a graphing utility to determine the number of
ways the committee can be formed.
38. Committee Selection Use a graphing utility to
determine the number of ways a committee of five
people can be formed from a group of 30 people.
39. Menu Selection A group of four people go out to
dinner at a restaurant. There are nine entrees on the
menu and the four people decide that no two will
order the same thing. How many ways can the four
order from the nine entrees?
40. Test Questions A student is required to answer any
9 questions from the 12 questions on an exam. Use a
graphing utility to determine the number of ways the
student can select the nine questions.
41. Basketball Lineup A high school basketball team
has 15 players. Use a graphing utility to determine
the number of ways the coach can choose the starting lineup. (Assume that each player can play any
position.)
42. Softball League Six churches form a softball
league. Each team must play every other team twice
during the season. What is the total number of league
games played?
D10
APPENDIX D
Counting Principles and Probability
43. Defective Units A shipment of 10 microwave
ovens contains 2 defective units. In how many ways
can a vending company purchase three of these units
and receive
(a) all good units?
(b) two good units?
(c) one good unit?
44. Job Applicants An employer interviews six people
for four job openings. Four of the six people are
women. If all six are qualified, in how many ways
can the employer fill the four positions if
(a) the selection is random?
(b) exactly two women are selected?
45. Group Selection Four people are to be selected
from four couples.
(a) In how many ways can this be done if there are
no restrictions?
(b) In how many ways can this be done if one person
from each couple must be selected?
46. Geometry Eight points are located in the coordinate plane such that no three are collinear. How
many different triangles can be formed having three
of the eight points as their vertices?
47. Geometry Three points that are not on a line determine three lines (see figure). How many lines are
determined by seven points, no three of which are on
a line?
Geometry In Exercises 48–51, find the number of
diagonals of the polygon. (A line segment connecting any two nonadjacent vertices of a polygon is
called a diagonal of the polygon.)
48. Pentagon
49. Hexagon
50. Octagon
51. Decagon
52. Relationships As the size of a group of people
increases, the number of relationships increases
dramatically (see figure). Determine the numbers of
different two-person relationships in groups of the
following numbers.
(a) 3 (b) 4 (c) 6 (d) 8 (e) 10 (f) 12
n=2
n=3
n=4
n=5