Civil Engineering Department: Engineering Statistics (ECIV 2005)

Engr. Yasser M. Almadhoun Page 1

Chapter 1: Probability Theory (Cont’d)

Section 1.7: Counting Techniques

Problem (01): A menu has five appetizers, three soups, seven main courses, six salad

dressings, and eight desserts.

(a) In how many ways can a full meal be chosen?

(b) In how many ways can a meal be chosen if either an appetizer or a

soup is ordered, but not both?

(Problem 1.7.4 in textbook)

Solution:

Problem (02): In an experiment to test iron strengths, three different ores, four different

furnace temperatures and two different cooling methods are to be

considered. Altogether, how many experimental configurations are

possible?

(Problem 1.7.5 in textbook)

Solution:

Problem (03): Four players compete in a tournament and are ranked from 1 to 4. They

then compete in another tournament and are again ranked from 1 to 4.

Suppose that their performances in the second tournament are unrelated to

Civil Engineering Department: Engineering Statistics (ECIV 2005)

Engr. Yasser M. Almadhoun Page 2

their performances in the first tournament, so that the two sets of rankings

are independent.

(a) What is the probability that each competitor receives an identical

ranking in the two tournaments?

(b) What is the probability that nobody receives the same ranking twice?

(Problem 1.7.6 in textbook)

Solution:

Problem (04): Twenty players compete in a tournament.

(a) In how many ways can rankings be assigned to the top five

competitors?

(b) In how many ways can the best five competitors be chosen (without

being in any order)?

(Problem 1.7.7 in textbook)

Solution:

𝑃𝑘𝑛 =

𝑛!

(𝑛 − 𝑘)!

𝑃520 =

20!

(20 − 5)!= 1,860,480

Civil Engineering Department: Engineering Statistics (ECIV 2005)

Engr. Yasser M. Almadhoun Page 3

𝐶𝑘𝑛 =

𝑛!

(𝑛 − 𝑘)! × 𝑘!

𝐶520 =

20!

(20 − 5)! × 5!= 15,504

Problem (05): There are 17 broken lightbulbs in a box of 100 lightbulbs. A random

sample of 3 lightbulbs is chosen without replacement.

(a) How many ways are there to choose the sample?

(b) How many samples contain no broken lightbulbs?

(c) What is the probability that the sample contains no broken

lightbulbs?

(d) How many samples contain exactly 1 broken lightbulb?

(e) What is the probability that the sample contains no more than 1

broken lightbulb?

(Problem 1.7.8 in textbook)

Solution:

𝐶𝑘𝑛 =

𝑛!

(𝑛 − 𝑘)! × 𝑘!

𝐶3100 =

100!

(100 − 3)! × 3!= 161,70

Civil Engineering Department: Engineering Statistics (ECIV 2005)

Engr. Yasser M. Almadhoun Page 4

𝐶𝑘𝑛 =

𝑛!

(𝑛 − 𝑘)! × 𝑘!

𝐶017 × 𝐶3

83 =83!

(83 − 3)! × 3!= 91,881

𝑃(𝐴) =𝐶3

83

𝐶3100 =

91,881

161,700= 0.5682

𝐶117 × 𝐶2

83 =17!

(17 − 1)! × 1!×

83!

(83 − 2)! × 2!= 57,851

𝑃(𝐵) =𝐶0

17 × 𝐶383 + 𝐶1

17 × 𝐶283

𝐶3100 =

91,881 + 57,851

161,700= 0.9260

Problem (06): (a) In how many ways can six people sit in six seats in a line at a

cinema?

(b) In how many ways can the six people sit around a dinner table eating

pizza after the movie?

(Problem 1.7.12 in textbook)

Civil Engineering Department: Engineering Statistics (ECIV 2005)

Engr. Yasser M. Almadhoun Page 5

Solution:

Problem (07): Repeat the previous problem with the condition that one of the six people,

Andrea, must sit next to Scott.

(a) In how many ways can six people sit in six seats in a line at a

cinema?

(b) In how many ways can the six people sit around a dinner table eating

pizza after the movie?

(c) In how many ways can the seating arrangements be made if Andrea

refuses to sit next to Scott?

(Problem 1.7.13 in textbook)

Solution:

−

Civil Engineering Department: Engineering Statistics (ECIV 2005)

Engr. Yasser M. Almadhoun Page 6

−

–

Problem (08): In how many ways can 7 people be seated at a round table if:

(a) they can sit anywhere?

(b) two particular people must sit next to each other?

(c) two particular people must not sit next to each other?

(Dr. Ramadan Al-Khatib’s webpage)

Solution:

Problem (09): A garage employs 14 mechanics, of whom 3 are needed on one job and, at

the same time, 4 are needed on another job. The remaining 7 are to be kept

in reserve. In how many ways can the job assignments be made?

(Problem 1.7.16 in textbook)

Civil Engineering Department: Engineering Statistics (ECIV 2005)

Engr. Yasser M. Almadhoun Page 7

Solution:

=𝑛!

𝑛1! × 𝑛2! × … × 𝑛𝑘!

=14!

3! × 4! × 7!

Problem (10): Five red balls, two white balls, and three blue balls are arranged in a row.

How many different arrangements are possible, knowing that all the balls

of the same colour are not distinguishable from each other?

(Dr. Ramadan Al-Khatib’s webpage)

Solution:

Problem (11): A quality inspector selects a sample of 12 items at random from a collection

of 60 items, of which 18 have excellent quality, 25 have good quality, 12

have poor quality, and 5 are defective.

(a) What is the probability that the sample only contains items that have

either excellent or good quality?

(b) What is the probability that the sample contains three items of

excellent quality, three items of good quality, three items of poor

quality, and three defective items?

Civil Engineering Department: Engineering Statistics (ECIV 2005)

Engr. Yasser M. Almadhoun Page 8

(Problem 1.7.18 in textbook)

Solution:

𝑃(𝐴) =𝐶12

43

𝐶1260 = 0.01096

𝑃(𝐴) =𝐶3

25 × 𝐶318 × 𝐶3

12 × 𝐶35

𝐶1260 = 0.002951

Problem (12): A box contains 40 batteries, 5 of which have low lifetimes, 30 of which

have average lifetimes, and 5 of which have high lifetimes. A consumer

requires 8 batteries to run an appliance and randomly selects them all from

the box. What is the probability that among the 8 batteries fitted into the

consumer’s appliance, there are exactly 2 low, 4 average and 2 high

lifetimes batteries?

(Problem 1.7.21 in textbook)

Civil Engineering Department: Engineering Statistics (ECIV 2005)

Engr. Yasser M. Almadhoun Page 9

Solution:

𝑃(𝐴) =𝐶2

5 × 𝐶430 × 𝐶2

5

𝐶840 = 0.0356

Problem (13): In a club of six men and six women, in how many ways can a committee

of four be selected if:

(a) The committee must consist of two men and two women?

(b) There must be at least one of each sex?

(c) The women must outnumber the men?

(Question 4: in Final Exam 2005)

Solution:

𝐶26 × 𝐶2

6 = 225

𝐶16 × 𝐶3

6 + 𝐶36 × 𝐶1

6 + 𝐶26 × 𝐶2

6 = 465

𝐶46 × 𝐶0

6 + 𝐶36 × 𝐶1

6 = 135

Civil Engineering Department: Engineering Statistics (ECIV 2005)

Engr. Yasser M. Almadhoun Page 10

Problem (14): In the library of the Islamic University of Gaza, there 6 statistic books and

6 structure books are available. In how many ways 4 books can be selected

from the two types of books if:

(a) The selected books must consist of two statistic books and two

structure books?

(b) There must be at least one of each type of the books?

(Question 4: (4 points) in Midterm Exam 2007)

Solution:

𝐶26 × 𝐶2

6 = 225

𝐶16 × 𝐶3

6 + 𝐶36 × 𝐶1

6 + 𝐶26 × 𝐶2

6 = 465

Problem (15): 5 different science books, 6 different history books, and 2 different statistic

books are to be arranged on a shelf. How many different arrangements are

possible if:

(a) (3 points) The books in each particular subject must all stand

together?

(b) (3 points) Only the science books must stand together?

(Question 3: (6 points) in Midterm Exam 2009)

Solution:

𝑛! = 3! = 3 × 2 × 1 = 6

5! × 8! = 4,838,400

Civil Engineering Department: Engineering Statistics (ECIV 2005)

Engr. Yasser M. Almadhoun Page 11

Problem (16): Five blue match sticks, three red match sticks, and two green match sticks

are to be arranged on a table. How many different arrangements are

possible if:

(a) (2 points) The same-colour sticks must stay together?

(b) (3 points) Only the blue sticks must stand together at the beginning?

(Question 2: (5 points) in Midterm Exam 2010)

Solution:

𝑛! = 3! = 3 × 2 × 1 = 6

5! × 5! = 14,400

Problem (17): Five different science books, six different history books, and two different

statistic books are to be arranged on a shelf.

(a) (2 points) How many different arrangements are possible if the

history books must stand together at the beginning.

(b) (4 points) Suppose that you need to select 6 books from the above-

mentioned book shelf to take home. In how many ways can the 6

books be selected for the following two cases:

(1) Your selection must consist of two science books, three

history books, and one statistic bock?

(2) Your selection must consist of at least two science books, at

least two history books, and exactly one statistic bock?

(Question 3: (6 points) in Midterm Exam 2011)

Solution:

Civil Engineering Department: Engineering Statistics (ECIV 2005)

Engr. Yasser M. Almadhoun Page 12

𝑛! = 6! × 7!

𝐶25 × 𝐶3

6 × 𝐶12 = 400

𝐶25 × 𝐶3

6 × 𝐶12 + 𝐶3

5 × 𝐶26 × 𝐶1

2 = 700