Thank you!

1. MABINI COLLEGES, INC.
High School Department
Flora A. Ibana Campus
Kapitan Isko St, Daet, Camarines Norte
A Detailed Lesson Plan in Mathematics 10
I – OBJECTIVES
A. Content Standards The learners demonstrate understanding of key concepts of combinatorics
and probability.
B. Performance Standards 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
Learning Competencies:
 The learner illustrates the permutation of objects. M10SP-IIIa-1
 The learner solves problems involving permutations. M10SP-IIIb-1
Specific Objectives:
At the end of the lesson, the learner should be able to:
1. defines concepts involving permutations;
2. applies different formula for finding the number of permutations;
and
3. cites instance where permutations are seen in real-life.
II – CONTENT Permutations
III – LEARNING MATERIALS
A. References
1. Teacher’s Guide Pages
2. Learner’s Materials
Pages
3. Textbook Pages/SLM Advance with Math pages 162-175.
4. Additional Materials
from LR
B. Other Learning Resources
IV – PROCEDURES Teacher’s Activity
Students’
Activity/Responses
A. Reviewing the
Previous Lesson or
presenting the new
Lesson
The teacher will show pictures and will ask the
students to identify the word/s being illustrated.
The teacher will ask the students to define or give
a brief description about the word/s.
ACTIVITY: SHOW KO, HULA MO!
Identify the terms with the help of the pictures
shown.
1. _ _ _ S _ _ _ _
The students will analyze
each set of photos and will
try to identify what these
pictures illustrate.
1. PASSWORD

2. 2. _ _ A _ _ _ G A _ _ _ N _ _ _ _ _ T
3. _ U _ _ _ _ D _ E _ _ _ _ S
Q: Based on the activity above, what does these
pictures illustrate?
Very Good!
So today, we’ll be going to discuss topic about
Permutations.
2. SEATING
ARRANGEMENT
3. JUMBLED LETTERS
A: The pictures illustrate
examples of Permutations.
B. Establishing a purpose
of the lesson.
Before we proceed to the lesson proper, let’s have first
our learning objectives.
The teacher will call a student to read the learning
objectives one at a time.
At the end of the lesson, you will be able to:
1. defines concepts involving permutations;
2. applies different formula for finding the
number of permutations; and
3. cites instance where permutations are seen in
real-life.
The students will raise their
hands to read.
At the end of the lesson, you
will be able to:
1. defines concepts
involving
permutations;

3. Thank you!
Now, lets proceed to our topic for today.
2. applies different
formula for finding
the number of
permutations; and
3. cites instance where
permutations are
seen in real-life.
C. Presenting examples
or instances of the new
lesson
Supposed we have three letters ABC, in how many
ways can we arrange these three letters, differently?
Give at least one arrangement.
The teacher will call a student to express their ideas
regarding the question.
Yes?
Very good! What else?
So, all in all, how many arrangements do we have?
Very good! Out of these 3 letters, we can have 6
possible arrangements.
The students will raise their
hands to answer.
Sir!
ACB
BAC
BCA
CAB
CBA
Sir! 6 arrangements including
ABC.
D. Discussing new
concepts and
practicing new skills 1
Now, based on that, we can define Permutations as?
The teacher will call a student to read the definition
flashed on the screen.
PERMUTATIONS:
A mathematical calculation of the number of ways a
particular set can be arranged. With permutations, the
order of the arrangement matters.
Thank you!
In order to find the number of permutations, we use
different counting techniques that we’ll discuss one by
one.
Are you ready class?
Okay, lets proceed.
So, there are plenty of ways of finding the number of
permutations.
The teacher will call a student to read the different
counting techniques flashed on the screen.
Students will raise their hands.
Sir!
PERMUTATIONS:
A mathematical calculation of
the number of ways a
particular set can be arranged.
With permutations, the order
of the arrangement matters.
Yes Sir!
The students will raise their
hands.

4. DIFFERENT COUNTING TECHNIQUES:
 Fundamental Principle of Counting or FPC
If there are m possible ways for an event to occur, and
n possible ways for another event to occur, then there
are m × n possible ways for both events to occur.
Thank you!
Example:
Ms. Lita Serrano is planning to organize a
symposium on proper waste management for
Senior High School students. How may possible
ways can she organize the symposium if she is
considering 3 school venues and 2 international
speakers?
First, we need to identify how many events occur.
Class, how many events occur in the given scenario?
Any idea?
Very good!
And what are those events?
Very Good!
By the use of the FPC, we can say that…
Event 1 is the school venues
Event 2 is the international speakers
Then, we can solve the number of possible ways by…
No. of possible ways = First Event x Second Event
No. of possible ways = m x n
No. of possible ways = 3 x 2
No. of possible ways = 6
Therefore, Ms. Serrano can organize the symposium in
6 possible ways.
Did you get it class?
Okay let’s proceed.
 Tree Diagram and Listing Method
Example:
A garment manufacturer has three sets of choices
for producing shirts:
Color = (red, blue)
Size = (small, medium, large)
Sleeve Cuts = (long, short)
 Can you list down the possible choices?
FPC states that, if there are m
possible ways for an event to
occur, and n possible ways for
another event to occur, then
there are m × n possible ways
for both events to occur.
Sir!
2 events Sir!
The number of venues and
the number of speakers.
Yes Sir!

5.  How many different kinds of shirts could the
manufacturer produce?
Illustrating the problem using the tree diagram will give
us…
Based on the illustration above, we can say that there
are 12 possible choices that the manufacturers can
produce. Is that right class?
But what are those kinds of shirts?
By listing method, try to identify the possible choices.
Who wants to try?
Yes?
Very good!
Did you get it class?
Okay, next one is…
The teacher will call a student to read the definition of
the Factorial Notation flashed on the screen.
 The Factorial Notation
A notation used to express the product of a series of
consecutive integers. If n is a positive integer, then “n
factorial” is defined as:
𝒏! = 𝒏 (𝒏 − 𝟏)(𝒏 − 𝟐) … … . (𝟏)
Yes Sir!
Sir!
Red – Small – Short
Red – Small – Long
Red – Medium – Short
Red – Medium – Long
Red – Large – Short
Red – Large – Long
Blue – Small – Short
Blue – Small – Long
Blue – Medium – Short
Blue – Medium – Long
Blue – Large – Short
Blue – Large – Long
Yes Sir!
The students will raise their
hands.

6. Note: Since the number of permutations of an empty set
is 1, meaning, an empty set can only be ordered one way,
thus, 0! = 1.
Example: Evaluate 4!
Since
𝒏! = 𝒏 (𝒏 − 𝟏)(𝒏 − 𝟐) … … . (𝟏)
Then,
𝟒! = 𝟒 (𝟒 − 𝟏)(𝟒 − 𝟐)(𝟒 − 𝟑)
𝟒! = (𝟒) (𝟑)(𝟐)(𝟏)
𝟒! = 𝟐𝟒
Therefore, 4! = 24.
Is that clear class?
Any question or clarification?
Yes Sir!
None Sir!
E. Discussing new
concepts and
practicing new skills 2
Okay, let’s now proceed to the different variations of
permutations.
Are you ready class?
Okay, let’s begin.
The teacher will discuss the different variations of
permutations one by one.
DIFFERENT VARIATIONS OF PERMUTATIONS
1. The Permutations of n objects taken r at a time.
𝑷(𝒏, 𝒓) =
𝒏!
(𝒏−𝒓)!
, 𝒏 ≥ 𝒓
Example: What is 𝑃 (5,2)?
𝑃(5,2) =
5!
(5 − 2)!
, 5 ≥ 2
𝑃(5,2) =
5!
3!
𝑃(5,2) =
5 · 4 · 3 · 2 · 1
3 · 2 · 1
𝑃(5,2) =
120
6
𝑷(𝟓, 𝟐) = 𝟐𝟎
2. The Permutations of n objects taken all at a
time.
Yes Sir!

7. 𝑷(𝒏, 𝒓) =
𝒏!
(𝒏−𝒓)!
, 𝒏 = 𝒓
Therefore, deriving the formula, it will become...
𝑷(𝒏, 𝒏) = 𝒏!
Example: What is 𝑃 (5,5)?
𝑃(5,5) = 5!
𝑃(5,5) = 5 · 4 · 3 · 2 · 1
𝑷(𝟓, 𝟓) = 𝟏𝟐𝟎
3. The Permutations of Distinct or Distinguishable
Objects
The number of distinguishable permutations, P, of
n objects where p objects are alike, q objects are
alike, r objects are alike, and so on, is
𝑷 =
𝒏!
𝒑! 𝒒! 𝒓! …
Example: Find the number of permutations of the
letters of the word PARALLEL.
The word PARALLEL has 8 letters. Assuming that
these letters are distinct, there are 𝑃 (8,8) or 8!
permutations. But, notice that 3 L’s are alike and
2 A’s are alike. The duplications are eliminated by
dividing 8! by the number of ways of arranging
the 3 L’s and 2 A’s which is 3!2!. It would be:
𝑃 =
(8,8)!
3! 2!
𝑃 =
8!
3! 2!
𝑃 =
8 · 7 · 6 · 5 · 4 · 3 · 2 · 1
3 · 2 · 1 · 2 · 1
𝑃 =
40, 320
12
𝑷 = 𝟑, 𝟑𝟔𝟎
Therefore, there are 3,360 distinguishable
permutations.
4. Circular Permutations
It is the arrangement of n objects in a circular order.
𝑷𝒏 = (𝒏 − 𝟏)!

8. where:
𝑷𝒏 = is the number of circular permutations
𝒏 = number of objects
Example: In how many ways can 5 people be seated
around a circular table?
Given:
𝒏 = 5
Required:
𝑷𝒏 = ?
Solution:
𝑃𝑛 = (𝑛 − 1)!
𝑃𝑛 = (5 − 1)!
𝑃𝑛 = 4!
𝑃𝑛 = 4 · 3 · 2 · 1
𝑷𝒏 = 𝟐𝟒
Therefore, there are 𝟐𝟒 circular permutations.
Class, is everything clear?
Well then, let’s proceed.
Yes Sir!
F. Developing mastery
(Leads to Formative
Assessment 3)
The teacher will ask the students to get their notebooks
and pen and to prepare for a drill.
Let’s have a DRILL!
A. Evaluate the following factorial notation.
1. 3! + 2!
2. (6-6)!
B. Using the tree diagram and listing method, find the
number of possible ways or choice of arrangement
on the given problems.
1. If Juan has 5 T-shirts, 3 pants, and 2 pairs of
shoes. How many possibilities can he dress
himself up for the day?
The students will prepare for
the drill.
Answers:
A.
1. 3! + 2!
= (3●2●1) + (2●1)
= 6 + 2
= 8
2. (6-6)!
= (0)!
= 0!
= 1

9. C. Solve the following problems using the different
formula for finding the number of permutations.
1. 𝑃 (6,5)
2. 𝑃 (8,8)
3. Find the number of permutations of the
letters/digits of the following:
a. REMEMBER
b. SCHOOL
4. There are 10 people in a dinner gathering. In
how many ways can the host (one of the 10)
arrange his guests around a dining table if:
a. They can sit on any of the chairs?
b. 3 people insist on sitting beside each
other?
c. 2 people refuse to sit beside each other?
B. Tree Diagram and Listing
Method
C. Solving:
1. 𝑃 (6,5) = 720
2. 𝑃 (8,8) = 40, 320
3. a. 1,680
b. 360
4. a. 3,628,800
b. 20,240
c. 80,640
G. Finding practical
applications of
concepts and skills in
daily life
Cite real-life instances where the concept of
permutations can be seen or applied. Explain.
Answer:
Thinking of 4-codes password
for cellphone lock. Computing
for the number of
permutations of numbers
from 0-9 in creating a 4-digit
password code will give us the
number of possible choices.
H. Making generalization
and abstraction
Based on our discussion, why is it important to study
the concept of permutations?
Answer:
To easily compute for the
number of possible choices
and/or combination.
I. Evaluating Learning Direction: Given the following situations/scenarios
below, find the number of permutations by
applying the appropriate counting techniques and
formula. SOLUTION IS REQUIRED.
1. In how many ways can 7 people arrange
themselves in a row for a picture taking?
2. Twelve runners join a race, in how many ways can
they be arranged as first, second and third placers?
3. Find the number of permutations of the letters
from the word STATISTICS.
Answers:
1. 5,040 possible ways
2. 1,320 ways
3. 50,400
4. 2,520 ways
5. a. 479,001,600 ways

10. 4. Five books in Mathematics, three in Literature, and
two in Science are to be arranged on a shelf that
has space just enough for these ten books.
Assuming that the books in the same subject are
identical, how many ways can they be arranged?
5. There are 13 people (including the host) in a dinner
gathering. In how many ways can the guests be
seated around a dining table if:
a. They can sit on any of the chairs?
b. 5 people insist on sitting beside each other?
c. 4 people refuse to sit beside each other?
b. 604,800 ways
c. 967,680 ways
Prepared by: Checked by:
LEVI R. OBUSAN JR. MIA ROSE V. DACILLO
Student Teacher Cooperating Teacher