The document contains a daily lesson log for a Grade 10 mathematics class. The lesson focuses on permutations. It outlines the objectives, content standards, learning competencies, content, learning resources, and procedures for the lesson. The procedures section provides examples and practice problems to help students understand permutations. It discusses concepts like permutation formulas and the fundamental counting principle. Students practice solving permutation problems involving objects, words, seating arrangements, and other examples from daily life. They work individually and in groups on word problems to find the number of permutations.

1. 1
= GRADE 1-12
DAILY LESSON
LOG
School Grade Level 10
Teachers Learning Area MATHEMATICS
Teaching Dates Quarter THIRD
MONDAY TUESDAY WEDNESDAY THURDAY FRIDAY
I. OBJECTIVES Objectives must be met over the week and connected to the curriculum standards. To meet the objectives necessary procedures must be
followed and if needed, additional lessons, exercises, and remedial activities may be done for developing content knowledge and competencies.
These are assessed using Formative Assessment strategies. Valuing objectives support the learning of content and competencies and enable
children to find significance and joy in learning the lessons. Weekly objectives shall be derived from the curriculum guides.
.
A. Content Standard The learner demonstrates understanding of the key concepts of combination and probability.
B. Performance Standard The learner is able to use precise counting technique and probability in formulating conclusions and
making decisions.
C. Learning
Competencies/Objectives
Write the LC code for each.
The learner illustrates
the permutation of
objects.
(M10SP-IIIa-1)
a. Illustrate the
permutation of objects.
b. List the possible ways
a certain task or activity
can be done
c. Appreciate
permutations as vital part
of one’s life.
The learner derives the
formula for finding the
number of permutations of
n objects taken r at a time.
(M10SP-IIIa-2)
a. Formulate the number of
permutation of n objects
taken r at a time.
b. Find the number of
permutation of n objects
taken at a time.
C. Appreciate permutations
as a vital part of one’s life.
The learner solves
problems involving
permutations
(M10SP-IIIb-1)
a. Solve problems
involving circular
permutations and
permutations with
repetitions.
b. Analyze each word
problem to identify the
given information
c. Value accumulated
knowledge as means
of new understanding
The learner solves
problems involving
permutations.
(M10SP-IIIb-1)
a. Solve problems
involving linear
permutations and
permutations taken r
at a time
b. Analyze each word
problem to identify
the given information
c. Value accumulated
knowledge as means
of new
understanding

2. 2
II. CONTENT Content is what the lesson is all about. It pertains to the subject matter that the teacher aims to teach in the CG, the content can be tackled
in a week or two.
Illustration of
Permutation
Permutation of n objects
taken at r time
Problem Solving Involving
Permutation
Problem Solving
Involving Permutation
III. LEARNING RESOURCES
A. References
1. Teacher’s Guide pages pp. 248-252 252-255 256 – 257 256 – 257
2. Learner’s Materials pages pp. 283-285 286-290 283 – 300 283 – 300
3. Textbook pages
Basic Probability and
Statistics, pp. 120-121
Elementary Statistics: A
Step by Step Approach, pp.
221-223
Basic Probability and
Statistics, pp. 120-121
Elementary Statistics: A
Step by Step Approach,
pp. 221-223
4. Additional Materials from
Learning Resource
(LR)portal
Worksheets and power
point
Worksheets and power
point presentation
Worksheets and power
point presentation
B. Other Learning Resource
https://onlinecourses.scie
nce.psu.edu/stat414/nod
e/29
http://www.analyzemath.
com/statistics/counting.ht
ml
http://www.mathsisfun.com
/data/basic-counting-
principle.html
http://www.math-
play.com/Permutations/per
mutations%20millionaire.ht
ml

3. 3
IV. PROCEDURES
A. Reviewing previous lesson or
presenting the new lesson
A. Preliminaries
Activating Prior
Knowledge
Erna invited her close
friends Chona, Mary
Grace and Emilie to her
18th birthday at Patio
Buendia in Amadeo. She
prepared a special table
with chairs placed in a
row to be occupied by
her three friends.
1. List all the possible
seating arrangements.
2. How many ways they
can be seated in a row?
3. Show another way/s of
finding the answer in
item 1.
Think-Pair-Share
Answer the
following with your
seatmate.
1. You have 3 shirts
and 4 pants. How
many possible
outfits can you
have?
2. There are 6 flavors
of ice-cream, and
3 different cones in
a grocery store.
How many orders
of ice cream can
you make?
The class will be divided
into 4 with uneven number
of members. Each group
will be asked to arrange
themselves in a circle. In
how many ways can this be
possible?
Drill
Compute the
permutations of the
following mentally.
1. P (4,2)
2. P (5,2)
3. P (6,1)
4. P (3,3)
5. P (7,4)

4. 4
B. Establishing a purpose for the
lesson
For personal password in
a computer account, did
you know why a shorter
password is “weak” while
the longer password is
“strong”?
Answer the following with
your seatmate.
Your task in this activity is
to think on how many
ways the following objects
can be arranged.
1.
Give real-life situations
where circular permutations
and permutations with
repeated elements.
Mr. Calix lost his ATM
card which can be opened
with a 4-digit password.
Should he be worried
overnight without
reporting the lost of his
card to the bank?

5. 5
2.
C. Presenting examples/Instances
of the new lesson
Permutation is
an arrangement of all or
parts of a set of objects
with proper order.
Permutations can be
determined by listing,
using table, tree
diagramming, and by
using the Fundamental
Counting Principle. FCP
is use to calculate the
total number of
permutations in a given
situation. The principle
may not tell what exactly
those permutations are,
but it gives the exact
number of permutations
there should be. The
FCP tells that you can
The different
arrangements which can
be made out of a given
number of things by taking
some or all at a time are
called permutation.
Let r and n be the
positive integers such that
1rn. Then the numbers
of all permutations of n
things taken at a time is
denoted by P(n,r) or nPr.
Let 1  r  n. Then the
number of all permutations
of n different things taken r
at a time is given by
P (n,r)=n!
(n-r)!
The number of
permutations of n things
taken r at a time is the
same as the number of
different ways in which r
Your mother made pickles,
gelatin, leche plan, ube jam,
sapin-sapin and graham.
You are to arrange the side
dishes and desserts in a
round table. Find the
circular permutation that you
can make.
One of the schools in the
province of Cavite will
conduct a beauty pageant
“Search for Binibining
Kalikasan”. For this year,
10 students join on the
said event. In how many
ways can second runner
up, first runner up and the
title holder be selected?
Solution:
Given: n = 10

6. 6
multiply the number of
ways each event can
occur.
Illustrative Example 1:
Suppose that you secure
your bike using a
combination lock. Later,
you forgot the 4- digit
code. You can only
remember that the code
contains the digits 1, 3,
4, and 7.
a. List all possible codes
out of the given digits.
b. How many possible
codes are there?
c. Use the Fundamental
counting principle to
check if the number of
permutations is correct.
Answer:
a. Possible codes
containing the four digits
7, 4, 3, 1:
(The list must be made
systematically to ensure
completeness.)
place in a row can be filled
with n different things.
The first place can be
filled up by any one of
these n things. So. Tthere
are n ways of filling up the
first place.
We are left with (n-1)
things. So, there are (n-1)
ways of filling the second
place.
Now, we are left with
n-2 things. So there are n-
2 ways of filling up the
third place.
By the fundamental
principle of counting, the
number of ways of filling up
the first three places is
n(n-1)(n-2).
Continuing this
manner, the rth place can
be filled up with any of
these n-(r-1) things. So
there are n-r+1 ways of
filling up the rth place.
Thus, the total number
of ways is
P(n,r) = n(n-1)(n-2)…(n-r+1)
=n(n-1)(n-2)…(n-r+1)((nr)…..3.2.1
(n-r)(n-r-1)….3.2.1
= n!
(n-r)!
Given: n = 6
Solution: P = (n – 1)!
= (6 – 1)!
= 5!
= 120
There are 120 ways to
arrange the side dishes and
desserts in a round table.
students
r = 3 winners
( )
( )
There are 720 ways to
select top three winners.

7. 7
1347 3147 4137 7134
1374 3174 4173 7143
1437 3417 4317 7314
1473 3471 4371 7341
1734 3714 4713 7413
1743 3741 4731 7431
b. There are 24 possible
outcomes.
c. Using the
Fundamental Counting
Principle:
1st digit 2nd digit 3rd digit
4th digit
4 choices × 3 choices
× 2 choices × 1 choice
= 24
Illustrative Example 2:
In how many ways can
Aling Rosa arrange 6
potted plants in a row?
Using the Fundamental
Counting Principle
Let N = number of
possible arrangements of

8. 8
the plants
N = (6) (5) (4) (3) (2)
(1)
N = 720 because there
are 6 choices for the 1st
position, 5 choices left
for the 2nd position, 4
choices for the 3rd, and
so on.
D. Discussing new concepts and
practicing new skills # 1
Complete the table
below:
Do you want to be a
Millionaire? Let’s Play!
Permutation Millionaire!
You have to answer every
question for 10 seconds.
Every correct answer has a
corresponding point. The
highest score a student
can earn will be an
additional point to become
a millionaire.
1. In how many ways can
three runners line up on
the starting line?
A. three B. Nine
C. Six D. Five
2. In how many ways can 4
books be arranged in a
shelf?
A. 24 B. 12
C. 8 D. 4
THINK-PAIR-SHARE
How many arrangements
can be made from the word
TAGAYTAY?
Solution:
let T equals n1.
A equals n2
G equals n3
Y equals n4
n= _____ n1 = _____
n2 = _____ n3 = _____
n4 = _____
=
_____________
The word TAGAYTAY can
be arranged into _______
ways.
THINK-PAIR-SHARE
Analyze the given
problem.
In how many ways can a
coach assign the starting
positions in a basketball
game to nine equally
qualified men?

9. 9
3. In how many ways can a
scoop of chocolate, a
scoop of vanilla and one of
strawberry be arranged on
an ice cream cone?
A. Six B. Nine
C. Ten D. Three
4. A class has 10 students.
How many choices for a
president and a vice-
president are possible?
A. 90 B. 1000
C. 100 D. 10,000
5. A couch can hold five
people. In how many ways
can five people sit on a
couch?
A. 120 B.125
C. 150 D.100
E. Discussing new concepts and
practicing new skills # 2
How did you determine
the different possibilities
asked for in the given
situations?
What mathematics
concept or principle did
you use to determine the
exact number of ways
asked in each activity?
How was the principle
applied?
Using the numbered heads
together answer the
following.
Find the number of
permutations of the letters
in the word PAPAYA .
1. How did you find the
activity?
2. What concepts of
permutations did you use to
solve the problem?
3. How did you apply the
principles of permutation in
solving the problem?
4. Can you cite other real-life
problems that can be solved
using permutation?
1. How did you find the
activity?
2. What concepts of
permutations did you use
to solve the problem?
3. How did you apply the
principles of permutation
in solving the problem?
4. Can you cite other real-
life problems that can be

10. 10
solved using permutation?
F. Developing mastery
(leads to Formative Assessment 3)
Solve the following
problems individually.
1. In how many ways can
you place 9 different
books on a shelf if there
is enough space for only
five books? Give 3
possible ways.
2. In how many ways can
5 people arrange
themselves in a row for
picture taking? Give 3
possible ways.
3. An apartment has 7
different units. There are
seven tenants waiting to
be assigned. In how
many ways can they be
assigned to the different
units? Give 3 possible
ways?
Answer the problem
individually.
How many
permutations does
each word have?
1. KURBADA
2. PALIKO
3. TUWID
Solve the following
problems
1. In how many ways
can 5 different
plants be planted in
a circle?
2. There are 4 copies
of Mathematics
book, 5 copies of
English book and 3
copies of Science
book. In how many
ways can they be
arranged on a
shelf?
Solve the following
problems.
1. Two raffle tickets
are drawn from
20 tickets for the
first and second
prizes. Find the
number of sample
points in the
sample spaces.
2. A teacher wants
to assign 4
different tasks to
her 4 students. In
how many ways
can she do it?
G. Finding practical application of
concepts and skills in daily living
Solve the following
problems individually.
1. In how many ways can
you place 9 different
books on a shelf if there
is enough space for only
five books? Give 3
possible ways.
2. In how many ways can
Group activity: In a
worksheet try to answer
the following using strips of
paper.
Directions: Find
the number of
permutations. Use the
(The students will be
working in groups and will
be presenting their output in
class.)
Solve the following
problems.
1. In how many ways can
4 students be seated at
around table?
(The students will be
working in groups and will
be presenting their output
in class.)
Solve the following
problems.
1. How many different
ways can a president
and a vice-president

11. 11
5 people arrange
themselves in a row for
picture taking? Give 3
possible ways.
3. An apartment has 7
different units. There are
seven tenants waiting to
be assigned. In how
many ways can they be
assigned to the different
units? Give 3 possible
ways?
formula and concepts you
learn from this lesson.
1. MALAYA
2. MAMAYA
3. MAMA
2. How many
arrangements can be
made from the word
CALCULATOR?
3. Find the number of
different ways that a
family of 6 can be
seated around a circular
table with 6 chairs.
4. How many
distinguishable
permutations are
possible with all the
letters of the word
ELLIPSES?
be selected for
classroom officers if
there are 30 students
in a class?
2. How many ways can
10 students line up in
a food counter?
3. In how many different
ways can 5 bicycles
be parked if there are
7 available parking
spaces?
4. In how many different
ways can 12 people
occupy the 12 seats in
a front row of a mini-
theater?
H. Making generalizations and
abstractions about the lesson
A permutation is an
arrangement of all or part
of a set of objects with
proper regard to order.
We determine the
different permutations by
listing. We also use
table, tree diagram and
as well as the
Fundamental Counting
Principle.
Remember: Permutation
is an arrangement, listing,
of objects in which the
order is important.
In general, when we are
given a problem involving
permutations, where we
are choosing r members
from a set with n members
and the order is important,
the number of
permutations is given by
Permutation with
Repeated Elements. The
number of distinct
permutation of n objects of
which n1 are one of a kind,
n2 of second kind, nk of a kth
kind is
where n1+ n2+ n3+…. = n
Circular Permutation.
When things are arranged in
places along a closed curve
Permutation is an
arrangement of n objects
taken in a specific order.
Linear Permutation. The
number of permutations of
n distinct of distinct
objects is n!
Factorial Notation. n! is
the product of the first n
consecutive natural
numbers.
Permutation of n
elements taken r at a
time

12. 12
the expression
nPr=n · (n - 1) · (n - 2) · …
·(n - r + 2) · (n - r + 1).
The first factor indicates
we can choose the first
member in n ways, the
second factor indicates we
can choose the second
member in n - 1 ways once
the first member has been
chosen, and so on.
or circle, in which any place
may be regarded as the first
or last place, they form a
circular permutation. Thus
with n distinguishable
objects we have (n-1)!
Arrangements. In symbol,
( )
( )
( )
where 0
≤ r ≤ n
I. Evaluating learning Study the following
situations. Identify which
situations illustrate
permutation. Then give
an example of possible
arrangements.
1. Determining the top
three winners in a
Mathematics Quiz Bee.
2. Choosing five group
mates for your
Mathematics project.
3. Three people posing
for a picture.
4. Assigning 4 practice
teachers to 4 different
Quiz
Answer each permutation
problem completely.
1. In how many ways
can 10 people line
up at a ticket
window of a
cinema hall?
2. Seven students
are contesting
election for the
president of the
student union. In
how many ways
can their names
be listed on the
ballot paper?
Solve the following
problems.
1. A man flips ten coins
among his ten children. The
coins are two one-centavo
coins, three five-centavo
coins, and five twenty-five
centavo coins. If each item
is to get one coin, in how
many ways can the children
share the coins?
2. A bracelet needs 10
chains of different colors. In
how many ways can the
chains be arranged or
Solve the following
problems.
1. A store manager
wishes to display 8
different brands of
shampoo in a row.
How many ways can
this be done?
2. Mar, Marlon, Marvin,
Martin and Marco
decided to go to SM
Dasmarinas. Each of
them has their own
motorcycle. Upon
arriving at the parking
lot, there are 7

13. 13
grade levels.
5. Picking 2 questions
from a bowl.
3. There are 3 blue
balls, 4 red balls
and 5 green balls.
In how many ways
can they be
arranged in a row?
joined to form a bracelet? available parking
spaces. In how many
different ways can
their motorcycle be
parked?
J. Additional activities for application
or remediation
1. Follow-up: How many
numbers consisting of 3
digits can be made from
1, 2, 3, 4, 5, and 6 if
a. Repetition is
allowed
b. Repetition is not
allowed
2. Study permutation of n
objects taken r at a time.
A. Follow-up.
Find the
permutation of the
following.
1. PACKAGE
2. MOUNTAIN
3. SCOUT
B. Study permutation
with repetition.
1. Follow-up
It is in international
summits that major world
decisions happen. Suppose
that you were the overall in
charge of the seating in an
international convention
wherein 12 country-
representatives were
invited. They are the prime
ministers/presidents of the
countries of Canada, China,
France, Germany, India,
Japan, Libya, Malaysia,
Philippines, South Korea,
USA, and United Kingdom.
1. If the seating
arrangement is to be
circular, how many seating
arrangements are possible?
2. Study : Combination
a. Differentiate
combination from
permutation
c. Give real-life situations
where combination can
1. Follow-up
In how many ways
can a jack, a queen and a
king be chosen from a
deck of 52 cards?
2. Study : Circular
Permutation and
Permutation with
Repetition
a. Give the formula for
circular permutation
and permutation with
repetition.
b. Give real-life
situations where
circular permutation
and permutation with
repetition can be
applied.

14. 14
be applied.
1. REMARKS
2. REFLECTION Reflect on your teaching and assess yourself as a teacher. Think about your students’ progress this week. What works? What
else needs to be done to help the students learn? Identify what help your instructional supervisors can provide for you so when
you meet them, you can ask them relevant questions.
A. No. of learners who earned 80%
in the evaluation
B. No. of learners who require
additional activities for
remediation who scored below
80%
C. Did the remedial lessons work?
No. of learners who have caught
up with the lesson
D. No. of learners who continue to
require remediation
E. Which of my teaching strategies
worked well? Why did these
work?
F. What difficulties did I encounter
which my principal or supervisor
can help me solve?
G. What innovation or localized
materials did I use/discover
which I wish to share with other
teachers?