W3
Learning Area
Quarter
Mathematics
Third
I. LESSON TITLE
II. MOST ESSENTIAL LEARNING
COMPETENCIES (MELCs)
III. CONTENT/CORE CONTENT
IV. LEARNING PHASES
A. Introduction
Suggested
Time Frame
20 minutes
Grade Level
Date
10
COMBINATION FORMULA
Derives the formula for finding the number of combinations of n objects taken r
at a time M10SP-IIId-1
Number of Combinations
Learning Activities
At the end of this lesson, you are expected to:
1. derive the combination formulas by applying the concepts learned
about permutation and FCP (Fundamental Counting Techniques);
2. organize the acquired knowledge from the previous lesson about
illustrating combinations; and
3. find the number of combinations of a set of objects by applying the
formula.
The lesson for this week is about deriving the Combination Formula.
From the previous lesson, you were able to learn the illustration of a
combination and differentiate permutation from combination.
Can you still remember the difference between permutation and
combination? In permutation, the order is important while in combination, order
is not important.
Are you now ready to learn more about combinations? Last week, we recalled
not only permutation but also the FCP (Fundamental Counting Principle). These
two are essential concepts in understanding combinations.
COMBINATION is an arrangement of n objects with no repetition and order is
not important. You learn to find the number of combinations of objects by
illustrating.
B. Development
60 minutes
Let us recall the difference between permutation and combination, again by
giving examples.
Deriving Combination Formula
Example 1. Suppose in your TLE subject, you are
assigned to make a menu, and one of the lists is SALAD.
If you decide to make a special chicken salad, you
need to add pineapple, apples, and grapes. You buy
these fruits from the market of San Pablo City. How
many ways are you going to arrange these fruits while
you are preparing them as ingredients? How can you
derive the formula for finding the number of combinations more systematically?
If the order is important then we have:
.
Based on the illustration, there are six ways if the order is important. This is
PERMUTATION.
IV. LEARNING PHASES
Suggested
Time Frame
Learning Activities
But if the order is not important you have, then you have the illustration bellow:
There is only one way. This is called COMBINATION.
Since you learned Permutation formula for the objects taken at all time is P(n, r),
where n = r
There are six possible ways if the order is important, however, let us have the
number of combinations. This does not consider the importance of order.
If by illustration the number of combinations is one, then we can conclude that
one is the answer for the number of combinations of objects taken all at a
time.
If the permutation of objects taken all at a time is nPn, then the combination of
objects taken at a time is nCn.
Therefore, for COMBINATION we have :
C(n , r) = C(n,n) or nCn = nCn =
𝑛𝑃𝑛
𝑛!
=1
Therefore : 3C3 = 1
Example 2. One day, a friend
of yours visited you and
fortunately, you are going to
visit your farm. He goes with
you there. Then, you think of
giving him a set of fruits, and
these are the combination of
fruits that you can find in your
orchard: buko, tamarind,
rambutan, and caimito. How
many combinations can you make if you select only three fruits out of four
choices?
By illustration:
buko - tamarind – rambutan
caimito – buko - rambutan
tamarind – rambutan – caimito
caimito – buko – tamarind
There are 4 combinations that you can make by illustrating.
IV. LEARNING PHASES
Suggested
Time Frame
Learning Activities
Let us derive the formula:
The number of different orders of four fruits taken three at a time is given
by:
There are 24 possibilities if order is significant, but we are looking for the
formula for finding the number of COMBINATION.
If P(4,3) is for permutation, then C(n,r) is for combination.
Since C(n, r) = C( 4,3) = 4 =
You know that 6 = (3)(2)(1) = 3!
Therefore, we have
If we let 4 = n; r = 3, then, the formula for finding the number of
combinations of n objects taken r at a time is:
Where: n > r > 0
C. Engagement
60 minutes
Learning Task 1: Flex that Brain! Find the missing value in each item.
1. C(8, 3) = ___
2. C(n, 4) = 15
3. C(9, 9) = ____
Learning Task 2: Choose Wisely, Choose Me! Solve the following problems
completely.
1.
2.
3.
If there are 12 teams in a basketball tournament and each team must
play every other team in the eliminations, how many elimination games
will there be?
If there are seven distinct points on a plane with no three are collinear,
how many different polygons can form?
How many different sets of five cards each can arrange from a
standard deck of 52 cards?
IV. LEARNING PHASES
Suggested
Time Frame
Learning Activities
Learning Task 3: Derive and Find! Derive the formula for finding the number of
combinations of the following situations:
1. Looking at the beautiful sky of a very peaceful night, you recognize the
Constellation Libra; it has shown six stars. Suppose you consider those stars as six
distinct points on a plane. How many polygons can be formed?
2. In how many ways can a MathSci Committee of five be formed from seven
Math lovers and five Science lovers if the committee must have three Math
lovers?
D. Assimilation
20 minutes
Level Up! Do this in your Journal Notebook.
1. Give two examples of situations in real life that involves permutations. In
each situation:
a. Formulate a problem.
b. Solve the problem.
2.
E.
Assessment
20 minutes
Explain how each particular may help you in formulating conclusions
and/or making decisions.
Solve this problem by applying the formula/s for combinations,
The 15th birthday party of your friend Cecilia will be held at Montelago Nature
Estates (San Pablo City). The motif will be unicorn and rainbow. You will be the
one to lead in decorating the function hall using balloons of different colors. A
box contains five peach balloons, seven pink balloons, six lavender balloons,
and four baby blue balloons. In how many ways can eight balloons be chosen
if there will be two balloons of each color?
VI. REFLECTION
Prepared by:
20 minutes
Maria Victoria V. Tiquis

The learners will write their personal insights about the lesson in their
notebook using the prompts below.
I understand that ___________________.
I realize that ________________________.
I need to learn more about __________.
Checked by: MA. FILIPINA M. DRIO