MATH 10
QUARTER 3 Week 5
NAME: ____________________________________ YR & SEC: _____________________
Competency
The learner solves problems involving permutations and combinations (M10SP-IIId-e-1)
To the Learners
Before starting the module, please set aside things and activities that will distract you while reading this
module and performing the activities. Read the instructions below to successfully enjoy the objectives of the kit.
Stay focus and enjoy!
1. Follow carefully the contents and instructions indicated in every page of this module.
2. Have your notebook beside you, so that you can write important details on it as well as computations of
the exercises.
3. You may use a calculator for large values of given.
4. Perform all provided activities in the module to have mastery of the concept.
5. Ask your parents/guardian to assist you in using this module.
Expectations
This module is designed to help you apply permutations and combinations in problem solving.
You will examine and determine whether the problem involves permutation or combination.
In this module, you should be able to:
1. determine whether a given problem involves permutation or combination
2. apply the formula in finding the permutation or combination of n objects taken r at a time
3. solve problems involving permutations and combinations
Pre-Test
Find out how much you already know about this module. Choose the letter that you think best
answers each of the following questions. Take note of the items that you were not able to answer correctly
and find the right answer as you go through this module.
1. From a group of 8 students, in how many ways can a president, a vice-president and a secretary be
chosen?
a. 256
b. 336
c. 392
d. 412
2. How many different arrangements of the letters of the word isosceles can be made?
a. 362 880
b. 181 440
c. 60 480
d. 30 240
3. Abby is selecting different 3 desserts from 10 possible choices displayed on the dessert cart at a
restaurant. How many selections are there?
a. 120
b. 240
c. 360
d. 720
4. A mother, a father, and their 4 children will dine in a restaurant with circular tables. In how many ways
can the family be seated in a table?
a. 24
b. 56
c. 120
d. 720
5. From a Grade 10 class of 28 students, five representatives are to be chosen for academic competition.
In how many ways can students be chosen to represent their class?
a. 12 285
b. 19 565
c. 49 140
d. 98 280
MATH 10 QUARTER 3 WEEK 5
1
6. Mr. Reyes has a vault with four-digit combination lock. He can set the combination himself. He can use
the digits 0 - 9. Each digit can be used only once. How many different combinations are possible?
a. 210
b. 1 260
c. 2 520
d. 3 024
7. To win a 6/42 lotto, a player chooses 6 numbers from 1 to 42. Each number can only be chosen once. If
all 6 numbers match the winning numbers, regardless of the order, the player wins. How many possible
winning numbers are there?
42!
42!
42!
42!
a.
b.
c.
d.
6!
36!
36!6!
42!6!
8. There are 8 scouts in a campsite. How many ways can the scouts sit around a campfire?
a. 40 320
b. 5 040
c. 720
d. 120
9. In how many different ways can the letters of the word MATHEMATICS be arranged?
11!
11!
11!
11!
a.
B.
c.
d.
2!
3!
2!3!
2!2!2!
10. There 12 basketball players. Each player can play any position. How many teams of 5 can be formed?
a. 792
b. 3 960
c. 19 008
d. 95 040
Looking Back
In previous lessons, we define permutation and combination. A permutation is an
arrangement, or listing, of objects in which the order is important. An arrangement of objects in which
the order is not important is called a combination.
The following diagrams give the formulas for permutation and combination.
Number of permutations (order matters)
of n things taken r at a time:
𝑃(𝑛, 𝑟) =
Number of combinations (order does not
matter) of n things taken r at a time:
𝑛!
(𝑛 − 𝑟)!
𝐶(𝑛, 𝑟) =
𝑛!
(𝑛 − 𝑟)! 𝑟!
Number of different permutations of n
objects where there are n1 repeated
items, n2 repeated items, … nk repeated
items:
𝑃=
𝑛!
𝑛1 ! 𝑛2 ! … 𝑛𝑘 !
Number of permutations of n things in a
circular arrangement:
𝑃 = (𝑛 − 1)!
Introduction
When solving problems involving permutations and combinations, it can be difficult to determine
between the two. Both permutations and combinations count the number of ways that (r) objects can be
taken from a group of (n) objects, but it is important to know that permutations are arrangements in which
the order matters, while combinations are selections in which the order does not matter.
The following are examples of problems involving permutations and combinations.
1. In a class, there are 12 students. In how many ways can three students be included in the top 3 position?
Solution:
The problem involves 12 students taken 3 at a time. Since the order matters in arranging the students in a
top 3 position, then the problem involves permutation.
Using the formula
MATH 10 QUARTER 3 WEEK 5
𝑛!
𝑃(𝑛, 𝑟) = (𝑛−𝑟)!
where n = 12 and r = 3,
2
𝑃(12, 3) =
12!
12!
12 ∙ 11 ∙ 10 ∙ 9!
=
=
= 1 320
(12 − 3)!
9!
9!
There are 1 320 ways that 3 students be included in top 3.
2. Jack is setting the code on a combination lock. If he wants to use to the numbers 14344 and has thought
of rearranging it, how may possible codes be there?
Solution:
Since the order matters, then the problem is a permutation. The problem involves 5 digits (1, 4, 3, 4, 4) where
there are repeated digits (the digit 4).
Using the formula 𝑃 =
𝑃=
𝑛!
𝑛1 !𝑛2 !…𝑛𝑘 !
where n = 5 and n1 = 3,
5!
5 ∙ 4 ∙ 3!
=
= 20
3!
3!
There are 20 possible codes.
3. In how many ways can 4 boys and 3 girls arrange themselves to sit in a round table?
Solution:
The problem involves 7 persons (4 boys and 3 girls) to be arranged in a round (circular) table.
Using the formula
𝑃 = (𝑛 − 1)!
where n = 7,
𝑃 = (7 − 1)! = 6! = 720
There 720 ways that the boys and girls can arrange themselves in a round table.
4. From among 8 students, 4 students will be chosen for a group dance presentation. How many possible
groups are there?
Solution:
The problem involves 8 students taken 4 at a time. In choosing the group member, the order is not important,
hence the problem involves combination.
Using the formula 𝐶(𝑛, 𝑟) =
𝐶(8, 4) =
𝑛!
(𝑛−𝑟)!𝑟!
where n = 8 and r = 4,
8!
8!
8 ∙ 7 ∙ 6 ∙ 5 ∙ 4! 1 680
=
=
=
= 70
(8 − 4)! 4!
4! 4!
4! 4!
24
There are 70 groups that can be made.
MATH 10 QUARTER 3 WEEK 5
3
Activity 1
Direction: Answer the following activity. Find the letter of the matching answer in the answer box
to decode the mathematician’s name.
Who is this female mathematician who inspired students to become mathematicians
themselves? Einstein said of her: “She discovered methods which have proven of
enormous importance in the development of the present-day generation of mathematicians.
She’s great in conceptual axiomatic thinking. Her genius ability is revealed in
her work with abstract concepts.’
N
504
T
10
Y
462
R
720
O
336
H
924
M Permutation
E
Combination
State if each situation involves permutation or combination.
1. Electing 4 candidates to a municipal planning
board from a field of 7 candidates
2. Ron is deciding 3 places to visit from a list of
10 places
3. Trying on PIN codes for an ATM card
4. There are 15 applicants to fill-in Clerical jobs
5. Ten club members need to choose a leader
and an assistant leader
Decide if each problem involves permutation or combination. Then solve each given problem.
6. Sam and Abby are planning to go on trips to
3 countries. There are 9 countries they would
like to visit. One trip will be one week long,
another two days, and the other two weeks.
How many trips can they plan?
7. A coach is lining up 6 starting players from 12
volleyball team members. In how many ways
can this be done?
8. Bob is listing his top 3 favorite songs from 10
different songs. In how many ways can he do
it?
9. A race has 8 competitors. In how many ways
can a first, second, and third-place finishers
be chosen?
MATH 10 QUARTER 3 WEEK 5
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10. In how many ways can you choose 2 different
desserts from a menu that offers ice cream,
cake, salad, leche flan and chocolate?
11. A company is hiring 5 engineers. If 11 people
apply for the job, how many different groups
of engineers can the company hire?
The mathematician’s name:
1
3
5
11
6
9
2
10
7
4
8
Activity 2
Direction: Use your knowledge of permutations and combinations to solve each problem in the box.
Use your answers to navigate through the maze. Color the correct path from start to end. Use a separate sheet
of paper for your solutions.
Start!
1. There are 6 seats in the
front row of an auditorium. In
how many ways can 6
students arrange themselves
to sit in the front row?
720
4. The password for Gena’s
email account consists of 4
characters chosen from the
letters g, e, n, a. How many
arrangements are possible, if
the password has no repeated
characters?
74
336
2. How many ways can the
letters in the word
PARALLEL be arranged?
120
3 360
24
120
56
45
6. There 20 kittens in a pet
shop. If three kittens will be
given away for adoption,
how many are possible?
840
10
1140
8. Suppose that 9 points are
distributed on a plane, such that
9. How many arrangements
no three points are on the same
50
400
line. Form a quadrilateral by
of the letters of the word
selecting the vertices from these
STATISTICS are possible?
points. How many quadrilaterals
are possible?
3 024
60
240
126
10. In how many ways can
a boy pick 8 marbles from a
box of 10 different colored
marbles?
45
11. A couple wants to plant
some shrubs around a
circular walkway. They have
seven different shrubs. How
many different ways can the
shrubs be planted?
144
720
10
28
14. A company wants to hire a
Computer Programmer, a
Software Tester and a
Systems Engineer. There are
10 applicants for the job. How
many ways can the company
hire the applicants?
MATH 10 QUARTER 3 WEEK 5
35
5. A pizzeria is offering a
special four-topping pizza.
There are 7 different
toppings available. How
many different four-topping
pizza are possible?
7. In how many ways can a
coach choose three
swimmers from among five
swimmers?
13. How many ways can
you select 2 of 8 different
brands of shirt at a
department store?
45
3. Abby, Benjie, Cathy, Dave,
Ernest, Fred Gail, and Harry
have won 3 tickets to the
opera. They will randomly
choose 3 people from their
group to attend the opera.
How many outcomes are
possible?
50 400
40 320
120
12. A club of nine people
wants to elect three
officers: President, Vice
President and Secretary.
How many ways can they
elect the officers?
504
720
900
End!
☺
5
Answers
1. ________
2. ________
3. ________
4. ________
5. ________
6. ________
7. ________
8. ________
9. ________
10. ________
11. ________
12. ________
13. ________
14. ________
Remember
Permutations and combinations are important statistical concepts. With permutations, we
calculate the number of possible rearrangements of a set of items. With combinations, we count the number of
combinations we can ‘choose’ from a larger set of items.
When the order doesn’t matter, it is Combination.
When the order does matter, it is Permutation.
Check Your Understanding
Solve the following problems on permutations and combinations.
1. Mike needs 8 more classes to complete his degree. If he met the prerequisites of all the courses, how
many ways can he take 4 classes next semester?
2. The Blaise is the official newspaper of BSHS. The staff of the newspaper has 16 students. In how many
ways can an editor-in-chief and assistant editor-in-chief be chosen among the staff?
3. To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How
many different hands are possible?
4. John and Michael want to arrange the letters of the word MISSISSIPPI and PHILIPPINES respectively.
If they started at the same time and at the same rate, which of the two will finish first?
Post-Test
Choose the letter of the correct answers in each of the following questions. Write your answers
in a separate sheet of paper.
1. From a group of 8 students, in how many ways can a president, a vice-president and a secretary be
chosen?
b. 256
b. 336
c. 392
d. 412
MATH 10 QUARTER 3 WEEK 5
6
2. How many different arrangements of the letters of the word isosceles can be made?
b. 362 880
b. 181 440
c. 60 480
d. 30 240
3. Abby is selecting 3 desserts from 10 possible choices displayed on the dessert cart at a restaurant. How
many selections are there?
b. 120
b. 240
c. 360
d. 720
4. A mother, a father, and their 4 children will dine in a restaurant with circular tables. In how many ways
can the family be seated in a table?
b. 24
b. 56
c. 120
d. 720
5. From a Grade 10 class of 28 students, five representatives are to be chosen for academic competition.
In how many ways can students be chosen to represent their class?
b. 12 285
b. 19 565
c. 49 140
d. 98 280
6. Mr. Reyes has a vault with four-digit combination lock. He can set the combination himself. He can use
the digits 0 - 9. Each digit can be used only once. How many different combinations are possible?
b. 210
b. 1 260
c. 2 520
d. 5 040
7. To win a 6/42 lotto, a player chooses 6 numbers from 1 to 42. Each number can only be chosen once. If
all 6 numbers match the winning numbers, regardless of the order, the player wins. How many possible
winning numbers are there?
42!
42!
42!
42!
b.
b.
c.
d.
6!
36!
36!6!
42!6!
8. There are 8 scouts in a campsite. How many ways can the scouts sit around a campfire?
b. 40 320
b. 5 040
c. 720
d. 120
9. In how many different ways can the letters of the word MATHEMATICS be arranged?
11!
11!
11!
11!
b.
B.
c.
d.
2!
3!
2!3!
2!2!2!
10. There 12 basketball players. Each player can play any position. How many teams of 5 can be formed?
b. 792
b. 3 960
c. 19 008
d. 95 040
Reflection
Fill in the template for your insight.
My Reflection
What did I learn
from the module?
MATH 10 QUARTER 3 WEEK 5
What is the most
interesting part of
the lesson?
What are the
things that I
realized while I
am doing the
activities in the
module?
What skill did I
learn that will be
useful in the
future?
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