Republic of the Philippines

Department of Education Regional Office IX, Zamboanga Peninsula

Mathematics 10 Quarter 3 - Module 3:

Combination of Objects

Zest for Progress

Zeal of Partnership

10

Name of Learner: ___________________________

Grade & Section: ___________________________

Name of School: ___________________________

1

What I Need to Know

The module contains only one lesson:

Lesson 3 - Illustrate combination of objects.

Lesson 4 - Differentiate permutation from combination of n objects taken r

at a time.

In this module, you are expected to:

a. Define and illustrate combination of objects.

b. Differentiate permutation from combination of n objects taken r at a time.

c. Relate combination in real life situation.

What I Know

Choose the letter of the correct answer. Write your answer on the space provided.

_____ 1. What do you call to the selection of objects regardless of their order?

A. combination C. integration

B. differentiation D. permutation

_____ 2. How many ways can a code be formed from the digits 0 to 9 if a combination

lock must contain 5 different digits?

A. 15 120 B. 30 240 C. 151 200 D. 1 000 000

_____ 3. Which of the following situations does NOT illustrate combination?

A. Selecting 2 songs from 10 choices for an audition piece.

B. Fixing the schedule of a group of students who must take exactly 8 subjects.

C. Enumerating the subsets of a set

D. Identifying the lines formed by connecting some given points on a plane.

_____ 4. What is C (10, 4)?

A. 210 B. 200 C. 220 D. 230

_____ 5. From 10 participants in a virtual orientation of Reginal Writers in Mathematics

10, 4 will be chosen to lead in each group. In how many ways can the participants be

chosen?

A. 24 B. 210 C. 5,040 D. 24 720

2

What’s In

Directions: Draw a heart if the given situation is a permutation, draw a diamond

if it is NOT a permutation.

_____ 1. A company has 12 members in its board of directors who will elect a president,

vice president, secretary, and treasurer among themselves. In how many ways can this

be done?

______2. In how many ways can identical pencils be shared by pupils?

_____3. In how many ways can bookmarks be divided among bookworms?

_____4. In how many different ways may 8 numbers be arranged in a spinner?

_____ 5. There are 8 boys in a basketball team. In how many ways can the coach select

5?

What’s New

“Shake it to the right, shake it to the left”

In a Regional Encampment held at Makilala, North Cotabato, there are six

troop leaders. How many handshakes are there if each troop leader shakes hands

with all the others once?

Clipart-library.com

Guide Questions:

1. What kind of situation is being presented?

2. How are you going to answer this problem?

3. What formula are you going to use?

3

What is it

Troop Leader A: shakes hand with B, then with C, D, E, and F. the total handshakes

made by troop leader A is 5;

Troop Leader B: shakes hand with C, then D, E and F. B did not shake hand with A

since the handshake between A and B is the same as between B and A (Order is NOT

important).

Troop Leader C: made 3 handshakes with D, E, and F.

Troop Leader D: made two handshakes with E and F.

Troop Leader F; make 1 handshake with F.

Total Number of handshakes is 15.

Combination refers to the selection of objects where order is not important. That is,

changing the order of the objects does not create a new combination.

For instance, the 3 combinations of the 3 letters T, I, and N taken 2 at a time are:

TI, TN, and IN

TI and IN are considered one combination. Similarly, IN and NI and TN and NT are the

same combinations.

There is only one combination that can be made from the letters T, I, and N taken 3 at a

time. That is TIN

TIN, TNI, INT, NIT, ITN, and NTI are considered one combination.

The combination of n things or objects taken r at a time can be denoted by

C(n,r).

4

How does combination differ from permutation?

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a. Your locker code is 543, if you enter 435 it won’t open because it is a different

ordering. Try to observe the table below.

Basic Comparison Permutation Combination

Meaning

Permutation refers to

the different ways of

arranging a set of

objects in sequential

order.

A combination refers to

several ways of selecting items

from a large set of objects,

such that their order does not

matters.

Order Important

Order matters

Not important

Doesn’t matter

Denotes Arrangement Selecting

Question

How many different

arrangements can be

created from a given

set of objects?

How many different groups

can be chosen from a larger

group of objects?

Formula/Notation

nPr =

( )

P(n,r), Pn,r or

nCr =

( )

C(n,r), C n,r or

Order does matter

(Permutation)

Order doesn’t matter

(Combination)

543

543

534

435

453

345

354

5

b. Consider the number of permutations of the 4 marbles with colors Red, Green,

Blue and Yellow taken 3 at a time.

The 24 permutations are listed below:

24 Permutations

There are 24 permutations but only 4 combinations. This is illustrated below:

4 Combinations

Example 2

For each experiment, tell whether arrangement is important or not.

a. Choosing 3 students from 30 students who will represent the class in Math,

Science, and English contests, respectively.

b. Choosing 2 teachers from 5 teachers in mathematics who will teach the 2

sections in Grade 10.

c. Selecting 3 pizza toppings from 10 toppings suggested in the menu in

Greenwich.

d. Choosing 10 songs from a Grand Videoke lists to be played in a party.

Answers:

a. To represent the class in Math, Science, and English contests, 3 students out of

30 will be chosen. After the 3 students are chosen, they will be assigned to the

contests that they will join. Thus, arrangement is important.

b. After the 2 mathematics teachers are chosen, they will be assigned to the 2

sections in Grade 10. Thus, arrangement is important.

c. The order of the pizza toppings is not important as all the toppings will be placed

on the pizza crust. Thus, arrangement is not important.

d. After 10 songs are chosen from the playlist, they will be lined up for playing in a

party. The different line-ups are the different arrangements of the songs. Thus,

arrangement is important.

6 permutations

6 permutations

6 permutations

6 permutations

1

Combination

1

Combination

1

6

Example 3

a. How many ways can 8 students be seated for a selfie if only 4 seats are available?

VectorSector.com Walmart.com

Solution:

There are 8 students to choose from, first seat can be filled in 8 ways. The

occupant of the 2nd seat can be chosen from the 7 students. Thus, the second seat

can be filled in 7 ways. After the first two seats are filled, the third and fourth

seats can be filled using the same reasoning, in 6 and 5 ways, respectively.

P(n,r) =

( )

P(8,4) =

( )

P(8,4) =

= 70

b. Suppose you are given an ordinary deck of playing cards. In how many was can 5

cards be selected?

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The number of ways of selecting r objects taken from n

distinct objects where arrangement is not important.

C(n,r) =

( )

7

Solution:

Selecting 5 cards from a deck of cards is a problem on selecting 5 objects

taken from 52 distinct objects where arrangement is not important. Thus, n = 52, and r

= 5, and

C(n,r) =

)

52C2 =

)

52C2 =

=

= 2, 598, 960

c. RRJ apparel is on 50% sale. If you are to select 2 shirts and 2 jeans of which there

8 designs for shirts and 6 different kinds of jeans, how many combinations do you

have?

C(n,r) =

( ) C(n,r) =

( )

C(6,2) =

( ) C(7,2) =

( )

=

( ) =

( )

=

=

= 15 = 28

= 15 28

= 420

Number of combinations of

2 jeans out of 6.

Number of combinations of

2 shirts out of 8.

“and” means multiply

“or” mean add

8

What’s More

Tell me more!

Given the following experiments, tell whether order is important or not. Then solve each

problem.

1. Choosing 3 winners out of the 10 semifinalists in a certain beauty pageant for the

titles Grand Winner, First Runner-up, and Second Runner-up.

2. Choosing 3 distinct flavors of ice cream from 10 flavors and placing them on an ice

cream cone.

3. Choosing 15 out of 40 students to join in the school’s Mathematics club.

4. Selecting 5 out of 7 available fruits in making fruit salad.

5. Selecting 4 officers from parents who will be President, Vice-President, Secretary,

and Treasurer.

What I Have Learned

Fill me in the value!

“What is the smallest number that can be divided

exactly by all the numbers from 1 to 10?”

Direction: Solve first all the problems and follow the pattern in filling in the blank in the

table to arrive the answer on the question, “What is the smallest number that can be

divided exactly by all the numbers from 1 to 10?”

1. How many 4-digit numbers can be formed from the digits 1, 3, 5, 6, 8, 9 if no

repetition is allowed? 360

2. In a group of 8 darters, only the top 2 will be given an award. In how many ways

can the organizer select the top 2 players? 20, 160

2 5 2 0

A B C D

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3. Suppose There are 7 consonants and4 vowels, how many words of 3 consonant

and 2 vowels can be formed?210 4. In how many ways can 4 mango trees be planted in 6 holes dug by the residents of

Pangi? 15

PATTERN

A = hundreds place value of number 3 answer

B = ones place value of answer in number 4

C = ten thousands value of answer in number 2

D = ones place value in number 1

What I Can Do

It’s your time to shine

1. Your mother went to budgetwise to buy the following: coffee (C), milk (M), sugar

(S), and butter (B). Upon reaching the grocery department, she found out that her

money is just enough to buy any three of these items. In how many ways can she

select the three items? List the possible selections.

2. How many choices do you have in the 11-combo meal at Jollibee

if your money is only enough for 4 choices?

3. Olivia was allowed to pick one meat (beef, chicken) and one flavor (spicy, Not spicy)

for her shawarma at Turks, how many different combinations can she make?

Assessment

Choose the letter of the correct answer. Please write your answer on the space provided.

_____1. What is the expanded form of 10C2?

A. C(10, 2) =

( ) C. C(10, 2) =

( )

B. C(10, 2) =

( ) D. C(10, 2) =

( )

10

_____2. What term refers to several ways of selecting items from a large set of objects,

such that their order does not matter?

A. combination C. integration

B. differentiation D. permutation

_____ 3. Which of the following experiments will determine that order is NOT important?

A. Selecting the top 3 winners in Math Quiz Bowl.

B. Setting a 4-digit code in a vault.

C. Buying 3 out 7 designs of face mask.

D. Assembling a jigsaw puzzle.

_____4. What is the correct answer if you will solve C(5,2)?

A. 10 B. 60 C. 20 D. 120

_____5. How many ways of selecting 3 flavors of ice cream can you make if there are 6

flavors available?

A. 15 B. 20 C. 30 D. 40

_____6. How many ways can 5 cars be parked if there are 7 available parking spaces?

A. 1260 B. 1540 C. 2230 D. 2520

_____7. In how many combinations can Coco have in the 5 displays of chicken inasal

with unlimited rice?

A. 1 B. 5 C. 10 D. 120

_____8. How many different words can we make from the letters L, O, V, and E? this

assumes every possible combination is a word.

A. 2 B. 6 C. 12 D. 24

_____9. At the grocery section, Kyvhan wanted to select 2 diapers, 2 milks,

3 shampoos.

In how many different selections does he have if there 5 diapers and 4 milks and 5

shampoos displayed?

A. 30 B. 50 C. 80 D. 120

_____10. How many ways can Cheem invites 3 or more friends to her birthday party if

she has on 5 friends?

A. 5 B. 10 C. 16 D. 50

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Mathematics 10 QUARTER 3 – MODULE 3 ANSWER KEY

WHAT I KNOW WHATS IN ASSESSMENT

1. A 2. A 3. B 4. A 5. B

1. PERMUTATION 2. COMBINATION 3. COMBINATION 4. PERMUTATION 5. COMBINATION

1. C 2. A 3. C 4. A 5. B 6. D 7. A 8. D 9. A 10. C

References

Callanta, Melvin M.,Allan M. Canonigo, Arnaldo I. Chua, Jerry D. Cruz, et.al, 2015,

Mathematics Learner’s Module, edited by Maxima J. Acelajado, 5th floor Mabini

Building, Deped Complex Meralco Avenue, Pasig City, Rex Books Store, Inc.

Ponsones, Rigor B., Arlene A. Pascasio, Shirlee R. Ocampo, Regina M. Tresvalles, 2013,

Math Ideas and Life Applications, Abiya, Building 851 G. Araneta Avenue, 1113

Quezon City, Philippines

Development Team

Writer: Christine Grace P. Saludes

Zamboanga Sibugay National High School

Editor/QA: Danniel M. Manlang

Ivy V. Deiparine Pede I. Casing

Reviewer: Gina I. Lihao EPS – Mathematics

Illustrator:

Layout Artist:

Management Team: Evelyn F. Importante OIC- CID Chief EPS

Dr. Jerry C. Bokingkito OIC- Assistant SDS

Dr. Aurilio A. Santisas, CESE OIC- Assistant SDS

Dr. Jeanelyn A. Aleman, CESO VI OIC-Schools Division Superintendent