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14
Counting and
Probability
otteries and gaming are as old as history records. Homer described how Agamemnon had his soldiers cast lots to see who
would face Hector. In the Bible, Moses divided the lands among
the tribes by casting lots.
Today 36 of the 50 states have lotteries and they are prosperous.
Nationwide, state-run lotteries generate $26.6 billion in income
annually—sales minus commissions. Available proceeds—money left
over for the states after paying out prizes and paying for administration
of the games totals $10.1 billion.
But what are your chances of winning a lottery? In this chapter we
will study the basic ideas of counting and probability. We will find the
probability of winning a lottery when we do Exercises 43 and 44 in
Section 14.3.
L
716
(14-2)
Chapter 14
Counting and Probability
14.1 C O U N T I N G A N D P E R M U T A T I O N S
In this
section
●
●
The Fundamental Counting
Principle
Permutations
V6
V8
V6
V8
V6
V8
Red
Blue
Gray
Although the main topic of this chapter is probability, we first study counting. We
all know how to count, but here we will learn methods for counting the number of
ways in which a sequence of events can occur.
The Fundamental Counting Principle
A new car is available in 3 different colors with 2 optional engines, V6 or V8.
For example, you can get a red car with a V6 engine or a red car with a V8 engine.
If the car comes in red, blue, or gray, then how many different cars are available?
We can make a diagram showing all of the different possibilities. See Fig. 14.1.
This diagram is called a tree diagram. Considering only color and engine type, we
count 6 different cars that are available from the tree diagram. Of course, we can
obtain 6 by multiplying 3 and 2. This example illustrates the fundamental counting
principle.
FIGURE 14.1
Fundamental Counting Principle
If event A has m different outcomes and event B has n different outcomes,
then there are mn different ways for events A and B to occur.
The fundamental counting principle can also be used for more than two events.
E X A M P L E
helpful
1
hint
When counting the number
of different possibilities, it
must be clear what “different”
means. In one problem we
might count a hamburger and
fries differently from fries and
a hamburger because we
are interested in the order in
which they occur. In another
problem we will consider
them as the same because
they are the same meal.
E X A M P L E
2
The fundamental counting principle
At Windy’s Hamburger Palace you can get a single, double, or triple burger. You
also have a choice of whether to include pickles, mustard, ketchup, onions,
tomatoes, lettuce, or cheese. How many different hamburgers are available at
Windy’s?
Solution
There are 3 outcomes to the event of choosing the amount of meat. For each of the
condiments there are 2 outcomes: whether or not to include it. So the number of
■
different hamburgers is 3 2 2 2 2 2 2 2 384.
The fundamental counting principle
How many different license plates are possible if each plate consists of 2 letters
followed by a 4-digit number? Assume repetitions in the letters or numbers are
allowed and any of the 10 digits may be used in each place of the 4-digit number.
Solution
Since there are 26 choices for each of the 2 letters and 10 choices for each of
the numbers, by the fundamental counting principle the number of license plates is
■
26 26 10 10 10 10 6,760,000.
14.1
Counting and Permutations
(14-3)
717
Permutations
The number of ways in which an event can occur often depends on what has already
occurred. In Examples 1 and 2 each choice was independent of the previous
choices. In Example 3 we will count permutations in which the number of choices
depends on what has already been chosen. A permutation is any ordering or
arrangement of distinct objects in a linear manner.
E X A M P L E
3
calculator
close-up
You can use factorial notation
on a calculator or the calculator’s built-in formula for permutations to find P(10, 3).
Permutations of n objects
How many different ways are there for 10 people to stand in line to buy basketball
tickets?
Solution
For the event of choosing the first person in line we have 10 outcomes. For the event of
choosing the second person to put in line only 9 people are left, so we have only 9
outcomes. For the third person we have 8 outcomes, and so on. So the number of permutations of 10 people is 10 9 8 7 6 5 4 3 2 1 10! 3,628,800.
■
The number of arrangements of the 10 people in Example 3 is referred to as the
number of permutations of 10 people taken 10 at a time, and we use the notation
P(10, 10) to represent this number. We have the following theorem.
Permutations of n Things n at a Time
The notation P(n, n) represents the number of permutations of n things taken
n at a time, and P(n, n) n!.
Sometimes we are interested in permutations in which all of the objects are
not used. For example, how many ways are there to fill the offices of president,
vice-president, and treasurer in a club of 10 people, assuming no one holds more
than one office? For the event of choosing the president there are 10 possible
outcomes. Since no one can hold 2 offices, there are only 9 possibilities for vicepresident, and only 8 possibilities for treasurer. So the number of ways to fill the
offices is 10 9 8 720. We say that the number of permutations of 10 people
taken 3 at a time is 720. We use the notation P(10, 3) to represent the number of
permutations of 10 things taken 3 at a time. Notice that
10! 10 9 8 7 6 5 4 3 2 1
P(10, 3) 10 9 8 720.
7654321
7!
study
tip
Be sure to ask your instructor
what to expect on the final
exam. Will it be written in the
same format as the other
tests? Will it be multiple
choice? Are there any sample
final exams available? Knowing what to expect will decrease your anxiety about the
final exam.
In general, we have the following theorem.
Permutations of n Things r at a Time
The notation P(n, r) represents the number of permutations of n things taken
n!
r at a time, and P(n, r) (n r)! for 0 r n.
Even though we do not usually take n things 0 at a time, we allow 0 in the formula.
For example,
8!
P(8, 0) 1
8!
and
0! 1
P(0, 0) 1.
0! 1
718
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Chapter 14
E X A M P L E
4
Counting and Probability
Permutations of n things r at a time
Eight prize winners will be randomly selected from 25 people attending a sales
meeting. There will be a first, second, third, fourth, fifth, sixth, seventh, and eighth
prize, each prize being of lesser value than the one before it. In how many different
ways can the prizes be awarded, assuming no one gets more than one prize?
Solution
The number of ways in which these prizes can be awarded is precisely the number
of permutations of 25 things taken 8 at a time,
25!
25!
P(25, 8) 25 24 23 22 21 20 19 18
(25 8)! 17!
4.3609 1010.
WARM-UPS
■
True or false? Explain your answer.
1. If a manufacturer codes its products using a single letter followed by a
single-digit number (0–9), then 36 different codes are available. False
2. If a sorority name consists of 3 Greek letters chosen from the 24 letters in the
Greek alphabet, with repetitions allowed, then 24 23 22 different sorority
names are possible. False
3. If an outfit consists of a skirt, a blouse, and a hat and Mirna has 4 skirts,
5 blouses, and 4 hats, then she has 80 different outfits in her wardrobe.
True
4. The number of ways in which 4 people can line up for a group photo is 24.
True
5.
100!
98!
9900
True
P(100, 98) 9900 False
The number of permutations of 5 things taken 2 at a time is 20. True
P(18, 0) 1 True
The number of different ways to mark the answers to a 10-question multiplechoice test in which each question has 5 choices is 105. False
10. The number of different ways to mark the answers to this sequence of
10 Warm-up questions is 210. True
6.
7.
8.
9.
14. 1
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is a tree diagram?
A tree diagram shows the number of different ways that a
sequence of events can occur.
2. What is the fundamental counting principle?
According to the fundamental counting principle, if event A
has m different outcomes and event B has n different outcomes, then there are mn ways for events A and B to occur.
3. What is a permutation?
A permutation is an ordering or arrangement of distinct
objects in a linear manner.
4. How many permutations are there for n things taken r at a
time?
The number of permutations of n things taken r at a time is
n!
.
(n r)!
14.1
Solve each problem. See Examples 1 and 2.
5. A parcel delivery truck can take any one of 3 different
roads from Clarksville to Leesville and any one of 4 different roads from Leesville to Jonesboro. How many different routes are available from Clarksville to Jonesboro?
12
6. A car can be ordered in any one of 8 different colors,
with 3 different engine sizes, 2 different transmissions,
3 different body styles, 4 different interior designs, and
5 different stereos. How many different cars are available?
2,880
7. A poker hand consists of 5 cards drawn from a deck of 52.
How many different poker hands are there consisting of an
ace, king, queen, jack, and ten? 1,024
FIGURE FOR EXERCISE 7
8. A certain card game consists of 3 cards drawn from a poker
deck of 52. How many different 3-card hands are there
containing one heart, one diamond, and one club? 2,197
9. Wendy’s Hamburgers once advertised that there were 256
different hamburgers available at Wendy’s. This number
was obtained by using the fundamental counting principle
and considering whether to include each one of several
different options on the burger. How many different
optional items were used to get this number? 8
10. A pizza can be ordered with your choice of one of 4 different meats or no meat. You also have a choice of whether to
include green peppers, onions, mushrooms, anchovies, or
black olives. How many different pizzas can be ordered?
160
FIGURE FOR EXERCISE 10
Counting and Permutations
(14-5)
719
Solve each problem. See Examples 3 and 4.
11. Randall has homework in mathematics, history, art, literature, and chemistry but cannot decide in which order to
attack these subjects. How many different orders are
possible? 120
12. Zita has packages to pick up at 8 different locations. How
many different ways are there for her to pick up the
packages? 40,320
13. Yesha has 12 schools to visit this week. In how many
different ways can she pick a first, second, and third school
to visit on Monday? 1,320
14. In how many ways can a history professor randomly assign
exactly one A, and B, one C, one D, and six F’s to a class of
10 students? 5,040
15. The program director for an independent television station
has 34 one-hour shows available for Monday night prime
time. How many different schedules are possible for the
7:00 to 10:00 p.m. time period? 35,904
16. A disc jockey must choose 8 songs from the top 40 to play
in the next 30-minute segment of his show. How many
different arrangements are possible for this segment?
3.1008 1012
Solve each counting problem.
17. How many different ways are there to mark the answers to
a 20-question multiple-choice test in which each question
has 4 possible answers? 1.0995 1012
18. A bookstore manager wants to make a window display that
consists of a mathematics book, a history book, and an economics book in that order. He has 13 different mathematics
books, 10 different history books, and 5 different economics books from which to choose. How many different displays are possible? 650
19. In how many ways can the 37 seats on a commuter flight be
filled from the 39 people holding tickets?
1.01989 1046
20. If a couple has decided on 6 possible first names for their
baby and 5 possible middle names, then how many ways
are there for them to name their baby? 30
21. A developer builds houses with 3 different exterior styles.
You have your choice of 3, 4, or 5 bedrooms, a fireplace,
2 or 3 baths, and 5 different kitchen designs. How many
different houses are available? 180
22. How many different seven-digit phone numbers are available in Wentworth if the first 3 digits must be 286?
10,000
23. How many different seven-digit phone numbers are availbale in Creekside if the first digit cannot be a 0?
9,000,000
24. In how many different ways can a Mercedes, a Cadillac,
and a Ford be awarded to 3 people chosen from the 9 finalists in a contest? 504
720
(14-6)
Chapter 14
Counting and Probability
25. How many different ways are there to seat 7 students in a
row? 5,040
30. Use the fundamental counting principle to find the number
of subsets of the set a, b, c, d, e, f . 64
31. How many ways are there to mark the answers to a test that
consists of 10 true-false questions followed by 10 multiplechoice questions with 5 options each? 1 1010
32. A fraternity votes on whether to accept each of 5 pledges.
How many different outcomes are possible for the vote?
32
33. Make a list of all of the ways to arrange the letters in the
word MILK. How many arrangements should be in your
list? 24
FIGURE FOR EXERCISE 25
26. A supply boat must stop at 9 oil rigs in the Gulf of Mexico.
How many different routes are possible? 362,880
27. How many different license plates can be formed by using
3 digits followed by a single letter followed by 3 more
digits? How many if the single letter can occur anywhere
except last?
26,000,000, 156,000,000
28. How many different license plates can be formed by using
any 3 letters followed by any 3 digits? How many if we
allow either the 3 digits or the 3 letters to come first?
17,576,000, 35,152,000
29. Make a list of all of the subsets of the set a, b, c. How
many are there? 8
34. Make a list of all of the permutations of the letters A, B, C,
D, and E taken 3 at a time. How many permutations should
be in your list? 60
Evaluate each expression.
35. P(8, 3)
336
36. P(17, 4)
57,120
37. P(52, 0)
1
38. P(34, 1)
34
P(10, 4)
39. 4!
210
P(8, 3)
40. 3!
56
P(12, 3)
41. 3!
220
P(15, 6)
42. 6!
5005
14!
43. 3! 11!
364
18!
44. 17! 1!
18
98!
45. 95! 3!
152,096
87!
46. 83! 4!
2,225,895
14.2 C O M B I N A T I O N S
In this
section
In the preceding section we learned the fundamental counting principle and applied
it to finding the number of permutations of n objects taken r at a time. We will now
learn how to count the number of combinations of n objects taken r at a time.
●
Combinations of n Things r
at a Time
Combinations of n Things r at a Time
●
Permutations,
Combinations, or Neither
●
Labeling
Consider the problem of awarding 2 identical scholarships to 2 students among
4 finalists: Ahmadi, Butler, Chen, and Davis. Since the scholarships are identical,
we do not count Ahmadi and Butler as different from Butler and Ahmadi. We can
easily list all possible choices of 2 students from the 4 possibilities A, B, C, and D:
A, B
A, C A, D
B, C
B, D C, D
It is convenient to use set notation to list these choices because in set notation we
have A, B B, A. Actually, we want the number of subsets or combinations of
size 2 from a set of 4 elements. This number is referred to as the number of combinations of 4 things taken 2 at a time, and we use the notation C(4, 2) to represent it.
We have C(4, 2) 6.
If we had a first and second prize to give to 2 of 4 finalists, then
P(4, 2) 4 3 12 is the number of ways to award the prizes. If the prizes are
identical, then we do not count AB as different from BA, or AC as different from
CA, and so on. So we divide P(4, 2) by 2!, the number of permutations of the 2 prize
winners, to get C(4, 2).