The document outlines guidelines for proper conduct during online classes, emphasizing respectful communication, participation, and discipline. It also covers fundamental principles of counting, including tree diagrams to determine possible outcomes of events, showcasing various examples related to concert schedules, food choices, and student elections. The document concludes with practice problems that involve applying these counting principles in different contexts.

1. Good Morning! 

3. 1. Use proper greetings.
2. Use appropriate language and correct
spelling of words.
3. No eating during online sessions or classes.
4. Be respectful.
5. Always turn on your camera.
6. Mute your microphone.
7. Be participative.
8. Do not leave the class.
9. Observe discipline.
10. Proper ending.
Observe the following netiquettes:

4. CHECKING
OF
ATTENDANCE

5. Tree Diagram and
Fundamental
Principles of
Counting

6. Learning Targets:
At the end of the module, you will be able to:
• Determine the number of ways a compound
event may occur.
• Use a tree diagram in determining all possible
outcomes of a compound event.
• State the Fundamental Principles of Counting.
• Solve counting problems using Fundamental
Principles of Counting.

7. 7
Tree Diagram
Sarah Geronimo is a popular concert artist. Suppose
she is planning a concert tour in three cities – Manila, Cebu
and Davao. In how many ways can she arrange her tour
schedule?
If there is no restriction on the order of performances,
then Sarah may start in any one of the three cities. After the
first city is chosen, she may choose from the two remaining
cities as her second stop. The remaining city will her last
stop.

8. 8
The tree diagram below shows the possible tour
schedules.
Start
First Stop
Manila
Cebu
Davao
Second Stop
Cebu
Davao
Manila
Davao
Manila
Cebu
Third Stop
Davao
Cebu
Davao
Manila
Cebu
Manila

9. 9
When the combinations of items or a succession of events
are considered, each result is called outcome. An event is a
subset of all possible outcomes. A compound event occurs when
an event is composed of two or more outcomes, such as flipping a
coin followed by flipping another coin.
The possible tour schedules in the compound event above
are as follows:
1 Manila – Cebu – Davao
2 Manila – Davao – Cebu
3 Cebu – Manila – Davao
4 Cebu – Davao – Manila
5 Davao – Manila – Cebu
6 Davao – Cebu – Manila
There are 6
possible tour
schedules.

10. 10
Remember
When a task can be done in two or more
stages and each stage can be done in a
number of ways, tree diagrams help in
showing the possible choices and in
determining the number of ways that the
whole task can be done.

11. 11
Example 1
A food stall sells squid ball, fish ball, and kikiam. There are
also three choices for the sauce: sweet, spicy, or sweet and
spicy. How many different combinations are possible?
Start
Food Choice
Squid ball
Fish ball
Kikiam
Sauce
sweet
spicy
sweet n’ spicy
sweet
spicy
sweet n’ spicy
sweet
spicy
sweet n’ spicy
Outcomes
SB, sweet
SB, spicy
SB, sweet n’ spicy
FB, sweet
FB, spicy
FB, sweet n’ spicy
K, sweet
K, spicy
K, sweet n’ spicy

12. 12
There are 9
possible
combinations.

13. 13
Example 2
A coin is tossed thrice. Draw a tree diagram to illustrate the
possible outcomes.
Start
First Toss
Head (H)
Tail (T)
Second Toss
Head (H)
Tail (T)
Head (H)
Tail (T)
Third Toss
Head (H)
Tail (T)
Head (H)
Tail (T)
Head (H)
Tail (T)
Head (H)
Tail (T)
HHH
HHT
HTH
HTT
THH
THT
TTH
TTT
There are 8 possible
outcomes.
There are 8 possible
outcomes.

14. 14
Fundamental Principle
of Counting
Concert Tour Schedule
First Stop Second Stop Third Stop
No. of Possible
Ways
3 2 1 6
Order in a Foot Stall
Food Choice Sauce
No. of Possible
Ways
3 3 9

15. 15
Fundamental Principle
of Counting
Tossing a Coin Thrice
First Toss Second Toss Third Toss
No. of Possible
Ways
2 2 2 8

16. 16
Is there a way of obtaining the number of possible ways or
outcomes without drawing a tree diagram?
Your answer leads to this useful principle.
The Fundamental Principle of Counting
In a compound event in which the first event may
occur in 𝑛1 different ways, the second event may occur in
𝑛2 different ways, and so on, and the 𝑘𝑡ℎ event may occur
in 𝑛𝑘 different ways, the total number of ways the
compound event may occur is 𝑛1 ∙ 𝑛2 ∙ 𝑛3 ∙. . . 𝑛𝑘.

17. 17
Example 3
In a student council election, there were 4 candidates for
president, 3 candidates for vice-president, 3 candidates for
secretary, and 4 candidates for treasurer. In how many ways
can these offices be filled?
Solution:
The president can be chosen in 4 ways, the vice-president in
3 ways, the secretary in 3 ways, and the treasurer in 4 ways.
By the Fundamental Principle of Counting, the four offices
can be filled in
𝟒 ∙ 𝟑 ∙ 𝟑 ∙ 𝟒 = 𝟏𝟒𝟒 𝒘𝒂𝒚𝒔
Solution:
The president can be chosen in 4 ways, the vice-president in
3 ways, the secretary in 3 ways, and the treasurer in 4 ways.
By the Fundamental Principle of Counting, the four offices
can be filled in
𝟒 ∙ 𝟑 ∙ 𝟑 ∙ 𝟒 = 𝟏𝟒𝟒 𝒘𝒂𝒚𝒔

18. 18
Example 4
New license plates for cars in the Philippines come in 3
letters and 4 digits format (LLL – DDDD).
a. How many license plates in this format are possible?
b. Of these, how many will have all their letters and digits
distinct?
Solution:
a. There are seven positions to be filled, the first three by
letters and the last four digits.
By the Fundamental Principle of Counting, the total
number of license plates possible is
L L L D D D D

19. 19
L L L D D D D
26 26 26 10 10 10 10
Solution:
𝟐𝟔 ∙ 𝟐𝟔 ∙ 𝟐𝟔 ∙ 𝟏𝟎 ∙ 𝟏𝟎 ∙ 𝟏𝟎 ∙ 𝟏𝟎 = 𝟏𝟕𝟓 𝟕𝟔𝟎 𝟎𝟎𝟎 𝒘𝒂𝒚𝒔

20. 20
Example 4
New license plates for cars in the Philippines come in 3
letters and 4 digits format (LLL – DDDD).
a. How many license plates in this format are possible?
b. Of these, how many will have all their letters and digits
distinct?
Solution:
b. For all the letters and digits to be distinct, it means that no
repetition is allowed.
𝟐𝟔 ∙ 𝟐𝟓 ∙ 𝟐𝟒 ∙ 𝟏𝟎 ∙ 𝟗 ∙ 𝟖 ∙ 𝟕 = 𝟕𝟖 𝟔𝟐𝟒 𝟎𝟎𝟎
plates with no repeated letter or digit.
26 25 24 10 9 8 7

21. Activity

22. 22
Read the problems carefully. Then, solve the problem.
*Using Tree Diagram
1.) In a café, coffee can be ordered in three sizes – small (S),
medium (M), and large (L), and in four flavors – Cappuccino
(C), Machiatto (M), Americano (A), and Latte (L). In how
many ways can you order coffee?
2.) A die is rolled once and a coin is tossed. What are the
possible outcomes?
*Using Fundamental Principle of Counting
3.) A particular type of combination lock has 10 numbers on
it. How many sequences of 3 numbers can be formed to
open the lock?

23. 23
4.) New license plate for motorcycles in the Philippines come
in 2 letters and 5 digits format (LL – DDDDD).
a. How many license plates in this format are possible?
b. Of these, how many will have all their letters and
digits distinct?
5.) At Maria’s Pizza, you can order pizza with thin or thick
crust and with any of the four toppings. The pizzas come in
solo, large, family, and extra-large sizes.
a. How many types of crust are there?
b. How many choices of toppings are there?
c. How many sizes of pizza are there?
d. How many ways can you order a one-topping pizza?

24. 24
6.) How many 4-letter codes can be formed with the letters M,
N, O, P, Q, R and S
a. with repetition?
b. without repetition?
7.) How many 5-letter codes can be formed with the letters U,
V, W, X, Y, and Z
a. with repetition?
b. without repetition?