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Mathematics
Quarter 3 – Module 3
Solving Problems Involving
Permutations and Combinations

LEOMAR A. GALSIM, Writer

Teacher, Bayambang National High School
 What I Need to Know

In this module, you will deal with problems involving permutations and
combinations. The arrangement of the content follows the standard sequence
of the course. But the pacing in which you read and answer this module is
dependent on your ability.

Learning Competencies:
The learner solves problems involving permutations and combinations.
After going through this module, you should be able to:
1. differentiate situations that involve permutations and
combinations; and
2. solve problems that involve permutations and combinations.

What I Know

Let us determine how much you already know about solving problems
involving permutations and combinations. If you answer all the test items
correctly, you may skip studying the content of this module and proceed to the
next.

DIRECTION: Read and understand each item, then choose the letter of your
answer and write it on your answer sheet.

1. Which of the following tasks involves combinations?

A. completing a jigsaw puzzle C. picking up numbers for a lottery
B. arranging frames in a wall D. positioning pieces on chessboard
2. Evaluate 9P3.
A. 504 B. 378 C. 168 D. 84
3. Which of the following is equal to 8C3?
A. 8!/3! B. 8!/5! C. 8!/3!5! D. 8(7)(6)
4. Which of the following situations involves permutations?
A. A 3-member committee is to be formed from 8 councilors.
B. A student is listing elements of a set in his Math assignment.
C. A group of friends is choosing 6 out of 10 main dishes for an occasion.
D. A board of executives is arranging shows in 3 TV primetime slots.
5. Evaluate 9C2.
A. 72 B. 48 C. 36 D. 24
6. In how many ways can a barangay captain choose 3 out of 7 councilors to
attend a seminar?
A. 35 B. 70 C. 105 D. 210

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7. How many distinct handshakes are possible if each person of the 14
persons inside a room shakes hand with everyone else once?
A. 91 B. 182 C. 273 D. 364
8. Yesha has 14 modules to answer this week. In how many different ways can
she pick a first, second, third, and fourth module to answer on Monday?
A. 24024 B. 6006 C. 2184 D. 1001
9. How many lines can be drawn from 10 given points, no 3 of which are
collinear?
A. 30 B. 45 C. 90 D. 135
10. Eight students entered a regional quiz bee. In how many ways can the first,
second, and third placers be chosen?
A. 24 B. 56 C. 336 D. 512
11. There are 7 men and 6 women in a club. How many different 3-member
committees can be chosen if two of them are men?
A. 27 B. 28 C. 105 D. 126
12. A disc jockey must choose 3 songs from the top 10 to play in the next 10-
minute segment. How many different arrangements are possible?
A. 720 B. 240 C. 120 D. 30
13. In the new plate number scheme, a motorcycle’s plate contains 2 letters
and 5 numbers. Which of the following expressions determines how many
different plate numbers are possible?
A. 26(25)(10)(9)(8)(7)(6) C. 26(26)(9)(9)(9)(9)(9)
B. 26(26)(10)(9)(8)(7)(6) D. 26(26)(10)(10)(10)(10)(10)
14. Six horses are needed to pull a float in a parade. If there are 10 horses in
the stable, how many different teams of 6 can be selected?
A. 5040 B. 1260 C. 60 D. 210
15. A gymnasium has 9 doors. In how many ways can you enter and leave the
gymnasium using different doors?
A. 90 B. 81 C. 72 D. 36

What’s In

The following activity enables you to review the previous lesson. This will help
you check your mastery of the pre-requisite skills for this learning module.
Activity 1: What’s my value??
Evaluate the following expressions.
1. 6P6 6. 8C2
2. 9P3 7. 7C4
3. 10P4 8. 10C10
4. 12P0 9. 20C0
5. 50P1 10. 50C1

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 What’s New

One of the best programs in our province is the Provincial Scholarship for the
college education of poor but deserving students. The qualifier or candidate for
scholarship must belong to a family with an annual combined income of
parents/guardians of not more than one hundred fifty thousand pesos (Php
150,000).

This year, 180 slots were allotted for the top

qualifying examination takers, where each
grantee was entitled to receive Php 10,000
financial assistance every term or semester
(Php 20,000 final assistance for the whole
academic year). Meanwhile, 30 slots were
allotted for automatic scholars. These
included the valedictorians of institutions in
senior high school.

(a) Suppose 20 students are semi-finalists for three scholarships – one

for Php 50,000, one for Php 30,000, and one for Php 20,000. In how
many different ways can the scholarships be awarded?

(b) Suppose 20 students are semi-finalists for three Php 30,000

scholarships. In how many different ways can the scholarships be
awarded?
]

What Is It

There is a wide range of problems involving permutations and combinations.

How do you know when to use the permutation formula and when to use the
combination formula? The key to answering this question is understanding
what each formula counts.
Compare the two situations in the preceding section.

The answer to the question in situation (a) is the number of permutations of

20 students taken 3 at a time because the order is important in awarding the
three different scholarships. On the other hand, the answer to the question in
situation (b) is the number of combinations of 20 students taken 3 at a time
because order does not matter in awarding equal scholarships.
The following steps are the solutions:

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 (a) 20P3
𝑛!
nPr = (𝑛−𝑟)!
20! 20! 20(19)(18)17!
20P3 = (20−3)! = 17! = = 20(19)(18)
17!

20P3 = 6,840
Therefore, there are 6,840 ways to award the three different scholarships.
(b) 20C3
𝑛!
nCr =
𝑟! 𝑛−𝑟)!
(

20! 20! 20(19)(18)17! 20(19)(18)

20C3 = 3!(20−3)! = 3!17! = = = 20(19)(3)
3!17! 3(2)(1)

20C3 = 1,140
Therefore, there are 1,140 ways to award the three equal scholarships.

Note that in either situation we are choosing 3 from a group of 20. The only
difference is whether to count only the ways of choosing or to count also their
arrangements. Check out and study more problems that involve permutations
and combinations in the following examples with stepped out solutions.

Example 1 A committee of three is to be chosen from a faculty of 13

Math teachers. How many ways of choosing the committee are
there?
Solution:

The members of the committee have no particular positions, so this problem

involves combinations.
𝒏!
nCr =
𝒓!(𝒏−𝒓)!

𝟏𝟑! 𝟏𝟑! 𝟏𝟑(𝟏𝟐)(𝟏𝟏)𝟏𝟎! 𝟏𝟑(𝟏𝟐)(𝟏𝟏)

13C3 = 𝟑!(𝟏𝟑−𝟑)! = 𝟑!𝟏𝟎! = = = 𝟏𝟑(𝟐)(𝟏𝟏)
𝟑!𝟏𝟎! 𝟑(𝟐)(𝟏)

13C3 = 𝟐𝟖𝟔
Therefore, there are 286 possible ways to choose the committee.

Example 2 Doc, Grumpy, Happy, Sleepy, Bashful, Sneezy, and Dopey

decided to go to work whistling in a different order. How many
days can they go without repeating an order?
Solution:

This problem involves permutations because different orders are counted. Out
of 7 characters, all are taken at a time, so we use nPn.
nPn = 𝒏!

7P7 = 𝟕! = 𝟕(𝟔)(𝟓)(𝟒)(𝟑)(𝟐)(𝟏)
7P7 = 𝟓, 𝟎𝟒𝟎
Therefore, they can go to work whistling in different order for 5,040 days.

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Example 3 Lisa has 12 modules to answer this week. In how many
different ways can she pick a first, second, and third module to
answer on Monday?
Solution:
Note that in this problem the order first, second, and third are taken into
account. Thus, this problem involves permutations.
𝒏!
nPr =
(𝒏−𝒓)!

𝟏𝟐! 𝟏𝟐! 𝟏𝟐(𝟏𝟏)(𝟏𝟎)𝟗!

12P3 = (𝟏𝟐−𝟑)! = = = 𝟏𝟐(𝟏𝟏)(𝟏𝟎)
𝟗! 𝟗!

12P3 = 1320
Alternate Solution:
12P3 = 𝟏𝟐(𝟏𝟏)(𝟏𝟎
⏟ ) = 𝟏𝟑𝟐𝟎

3 factors

Therefore, there are 1320 different ways to do it.

Example 4 Seven friends Jennie, Kevin, Lisa, Maria, Nick, Olivia, and
Paul, leave a restaurant. Each person says good-bye to each of
the others with a fist bump. How many fist bumps are
needed?
Solution:
The fist bump made by Jennie and Kevin, for example, are the same with the
fist bump made by Kevin and Jennie. Different orders are not counted so this
involves combinations. It takes two persons to have a fist bump so 𝑟 = 2.
𝒏!
nCr =
𝒓!(𝒏−𝒓)!

𝟕! 𝟕! 𝟕(𝟔)𝟓! 𝟕(𝟔)
7C2 = 𝟐!(𝟕−𝟐)! = 𝟐!𝟓! = = (𝟐)(𝟏) = 𝟕(𝟑)
𝟐!𝟓!

7C2 = 𝟐𝟏
Therefore, the number of fist bumps is 21.

Example 5 Six points are in a plane such that no three are collinear, as shown.
How many lines can be formed through these points?

Solution:
A line named ̅̅̅̅
𝐴𝐵 is the same with line named ̅̅̅̅
𝐵𝐴. Thus, this is a combination
problem. Two points determine a line, so 𝑟 = 2.
𝒏!
nCr = 𝒓!(𝒏−𝒓)!
𝟔! 𝟔! (𝟔)(𝟓)𝟒! 𝟔(𝟓)
6C2 = 𝟐!(𝟔−𝟐)! = 𝟐!𝟒! = = (𝟐)(𝟏) = 𝟑(𝟓)
𝟐!𝟒!

6C2 = 𝟏𝟓
Therefore, 15 lines can be formed.

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 What’s More

Now it’s your time to try out the activities for independent practice. This is to
ensure that you develop the target skills you are expected to learn.

Activity 2: Make a Difference

Determine whether each of the following situations involves permutations or
combinations.
1. electing a president, a vice- 7. drawing triangles through
president, and a secretary coplanar points
2. forming a committee 8. awarding gold, silver, and
3. selecting teachers for a bronze medals
seminar 9. setting up seating
4. displaying frames on a wall arrangements
5. creating a passcode 10. awarding 500-peso
6. arranging segments for a consolation prizes
variety show

Activity 3: Permutation Problems:

Read, analyze, and solve the following problems.

1. How many ways are there to elect a president, a vice-president, a

secretary, and a treasurer from a group of 9 people?
2. An exhibition hall has eight doors. In how many ways can you enter
and leave the hall through different doors?
3. You choose a 4-letter password for your account, but the system tells
you that it is “weak”. So, you decided to add 3 numbers. How many
choices are possible?

Activity 4: Combination Problemss

Read, analyze, and solve the following problems.

1. Ten friends meet at a wedding, and each person greets each other with
a hug. How many hugs take place?
2. Seven points are in a plane so that no three of them are collinear.
a. How many lines can be formed through these points?
b. How many diagonals can be formed through these points?
3. A five-member committee is to formed from 10 juniors and 9 seniors.
a. In how many ways can the committee be formed, if there are no
restrictions?
b. In how many ways can the committee be formed if it must include 2
juniors and 3 seniors?

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 What I Have Learned

Great! You have reached this part of the module. Now, organize your
thoughts about the lesson learned and answer the following:

1. State the difference between permutation and combination by

constructing a sentence using the words in the box.

objects order regard selection without arrangement

a in of to an with

A permutation is _______________________________________________
while a combination is
_______________________________________________.

2. You want to secure your travel bag with a “combination” lock that has a
4-digit code and you want to put in a code that has no repeated
numbers. Which counting technique is involved in the problem,
permutations or combinations? Explain your reasoning.

3. In your own words, how do you determine whether a problem involves

permutations or combinations?

What I Can Do

In this portion, try to apply your new knowledge and skill in real life
situations or scenarios by answering and completing the following tasks.

1. A lottery which requires guessing 6 numbers out of 50 in any order

sells 25-peso tickets. A group of people decides to buy 12,000,000
tickets because the jackpot prize is Php350,000,000.
a. Can they ensure that they win with their tickets? Support your
answer.
b. Is their decision correct? Explain your reasoning.

2. Write one problem that involves permutations and another problem

that involves combinations. Solve each problem and explain how you
can use these situations in your life especially in formulating
conclusions and making decisions.

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 Assessment

Let us determine how much you have learned from this module.
DIRECTION: Read and understand each item, then choose the letter of your
answer and write it on your answer sheet.
1. Find the number of permutations of 14 objects taken 3 at a time.
A. 2002 B. 2184 C. 364 D. 728
2. Jennie has homework in Math, Art, English, Science, and TLE but she
cannot decide in which order to attack these subjects. How many
different orders are possible?
A. 5! B. 4! C. 25 D. 10
3. In how many ways can Willie Revillame award Php 50,000 each to 3
people chosen from the 𝑛 viewers in his show?
A. n!/3!(n-3)! B. n!/3! C. n!/(n-3)! D. n!/3
4. Evaluate 7P4.
A. 210 B. 840 C. 35 D. 28
5. Evaluate 7C4.
A. 28 B. 35 C. 210 D. 840
6. Which of the following situations involves combinations?
A. A refrigerator, a TV, and a stand fan are to be awarded in a raffle
draw.
B. Parents elect president, vice-president, secretary, treasurer, and
auditor for PTA.
C. A librarian arranges books in a shelf.
D. An SK chairman appoints 3 SK kagawads to buy school supplies.
7. During the pandemic, people are practicing physical distancing. A group
of friends meet up at a restaurant. They greeted each other with a fist
bump. If 21 fist bumps took place, how many friends were present?
A. 5 B. 6 C. 7 D. 8
8. Which of the following tasks involves permutations?
A. Awarding 3 equal scholarships. C. Choosing 4 beaches to visit
B. Picking up 6 numbers for a lottery D. Arranging 7 books in a shelf
9. Evaluate and simplify: 5C5 + 5C4 + 5C3 + 5C2 + 5C1
A. 31 B. 32 C. 22 D. 51
10. A gymnasium has 9 doors. In how many ways can you enter and leave
the gymnasium using different doors?
A. 90 B. 81 C. 72 D. 36
11. Determine the number of triangles that can be formed through 10 points
no three of which are collinear.
A. 45 B. 90 C. 360 D. 120

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12. A 4-member committee is to be formed from the officers of a club. In
how many ways can the committee be formed if the club has 11 officers?
A. 990 B. 330 C. 1980 D. 7920
13. Six horses are needed to pull a float in a parade. If there are 10 horses
in the stable, how many different teams of 6 can be selected?
A. 5040 B. 1260 C. 60 D. 210
14. In how many ways can the organizers of Bayambang Singkapital Poem
Writing Contest award Php 10,000, Php 7,000, Php 5,000 to 10 finalists?
A. 720 B. 360 C. 120 D. 30
15. As part of a simple celebration, a family of six went to a studio. They
want to have pictures with different orders. How many shots with
different orders can they choose from?
A. 720 B. 120 C. 60 D. 36

Additional Activities

Enrich your knowledge and skill of the lesson learned by doing the
additional activity below.

1. Twelve students have volunteered to help clean up a small oil spill.

The project director needs 3 bird washers, 4 rock wipers, and 5 sand
cleaners. In how many ways can these jobs be assigned to these 12
students?

2. Eight points are coplanar and no three of them are collinear. Find how
many polygons can be formed through these points. Complete the
table.

Triangles Quadrilateral Pentagon Hexagon Heptagon Octagon Total

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Answer Key

References
Callanta, Melvin M., Canonigo, Allan M., Chua, Arnaldo I., Cruz, Jerry D.,
Esparrago, Mirla S., Garcia, Elino S., et al. 2015. Mathematics 10 Learner’s
Module. First Edition. Philippines: Department of Education

Hall, Bettye C., and Fabricant, Mona 1995. Algebra 2 with Trigonometry.
New Jersey, USA: Prentice-Hall, Inc.

Dugopolski, Mark 2006. Algebra for College Students. Fourth Edition. New
York, USA: McGraw-Hill Companies, Inc.

Senk, Sharon L., Viktora, Steven S., et al. 1998. Functions, Statistics, and
Trigonometry. Second Edition. The University of Chicago School
Mathematics Project. Illinois, USA: Addison Wesley Longman, Inc.

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