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1. FUNDAMENTAL
COUNTING PRINCIPLES

2. FUNDAMENTAL COUNTING PRINCIPLES
If the first operation can be performed in
n₁ ways and the second operation in n₂
ways, then the entire experiment can be
performed in n₁ x n₂ ways.

3. EXAMPLE #1
In the experiment of tossing a coin and
rolling a die, how many elements can be
made? n (Coin) = 2
₁
n (Die) = 6
₂
2 x 6 = 12 elements

4. HEAD TAIL
1
2
3
4
5
6
1
2
3
4
5
6
(head, 1)
(head, 2)
(head, 3)
(head, 4)
(head, 5)
(head, 6)
(Tail, 1)
(Tail, 2)
(Tail, 3)
(Tail, 4)
(Tail, 5)
(Tail, 6)
There are 12 elements

5. EXAMPLE #2
How many elements are there in the
experiment of choosing a color from red,
blue and yellow, and tossing a coin?
n1 (Coin) = 2
n2 (Color) = 3
2 x 3 = 6 elements

6. Red (R) Blue (B) Yellow (Y)
Head (H) HR HB HY
Tail (T) TR TB TY
There are 6 elements

7. EXAMPLE #3
Suppose you can have pancake, cereal, or sandwich
for your breakfast and juice or milk for your drink.
How many choices do you have in all for your
breakfast?
n1 (Food) = 3
n2 (Drinks) = 2
2 x 3 = 6 choices

8. EXAMPLE #4
Daniel is planning to purchase a photo album. It
comes in three sizes, small, medium, and large; and
the cover comes in hard or soft bound. The pages can
be glossy or silk, and the print can be colored or plain
black and white. How many choices does he have for
the photo album?

9. n1 (Sizes) = 3 n3
(Pages) = 2
n2 (Cover) = 2 n4 (Print) =
2
3 x 2 x 2 x 2 = 24 Choices

10. PERMUTATION

11. PERMUTATION
An arrangement of a given set.
In the arrangement of n objects there are n operations
involved. The first operation involves choosing an item
for the first position; the second operation, choosing an
item for the second position and so until the nth
operation.

12. LINEAR PERMUTATION FORMULA
n! = n x (n-1) x … 1

13. EXAMPLE #1
Suppose Debbie, Anna, Roy and Emily will
line up to buy lunch in the school canteen.
How many ways can the four students be
in line?

14. n = 4
n! = 4!
4! = 4 x 3 x 2 x 1 = 24 ways

16. EXAMPLE #2
In how many ways can the letters of the
word “FAITH” be arrange?
n = 5
n! = 5!
5! = 5 x 4 x 3 x 2 x 1 = 120 ways

17. EXAMPLE #3
Alex, Alvin, Alyssa, Alfred, Alan, and Aljon are to
occupy the front seats of the auditorium. In how
many ways can they arrange their seats?
n = 6
n! = 6!
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 ways

18. CIRCULAR PERMUTATION FORMULA
(n - 1)!

19. EXAMPLE #1
A family with 6 members sit in a round table
for dinner. How many ways can the member of
the family be seated?
n = 6
(n – 1)! = (6 - 1)!
5! = 5 x 4 x 3 x 2 x 1 = 120 ways

20. EXAMPLE #2
In how many ways can 7 appetizers be
arranged on a circular tray?
n = 7
(n – 1)! = (7 - 1)!
6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 ways

21. PERMUTATION WITH REPETITION
OF ELEMENTS
𝒏!
(𝒏¿¿𝟏!)(𝒏𝟐!)..(𝒏𝒌 !)¿

22. EXAMPLE #1
In how many ways can 2 blue balls, 2 red balls,
and a yellow ball be arrange?
n = 5
n1 = 2
n2 = 2
=

23. EXAMPLE #2
In how many ways can the letters of the word
“ARRIVE” be arrange?
n = 6
n1 = 2
=

24. EXAMPLE #3
In how many ways can the numbers
“012314501” be arrange?
n = 9
n1 = 2
n2 = 3
=

25. PERMUTATION OF N OBJECTS
TAKEN R AT A TIME
❑𝑵 𝑷𝑹=
𝒏!
(𝒏 −𝒓 )!

26. EXAMPLE #1
How many two digits number can you make
from numbers 1, 2, 3, and 4?
n = 4
r = 2
4P2 = = = 12 ways

27. EXAMPLE #2
In how many ways can you arrange 6 books in
a bookshelf, if the capacity of the bookshelf is
only 4 books?
n = 6
r = 4
6P4 = =
= 360 ways

28. EXAMPLE #3
Brian, Brenda, Brix, Brandon, and Brylle are to
sit in the front seat of the auditorium, if there
are only 2 seats left, how many ways can they
choose and arrange 2 persons to sit in the
front seat of the auditorium?

29. EXAMPLE #3
n = 5 r =
2
5P2 = = = 20 ways

30. EXAMPLE #4
There are 7 members in a group. Three of
them are to be appointed as president, vice
president, and secretary. How many ways can
one choose the president, the vice president,
and the secretary from the group?

31. EXAMPLE #4
n = 7 r =
3
7P3 = = = 210 ways

32. ANSWER THE
FOLLOWING

33. PROBLEM #1
In how many ways can the letters
of the word “COURAGE” be
arranged?

34. PROBLEM #2
A family consist of 9 members, In
how many ways can they arrange
their seat in a round table?

35. PROBLEM #2
A family consist of 9 members, In
how many ways can they arrange
their seat in a round table?

36. PROBLEM #3
Suppose you have 7 pens, 3 of them
are color blue, 2 are color red and 2
are color black, in how many ways can
you arrange the pens in a pen case?

37. PROBLEM #4
How many arrangement can you
make if there are 8 persons to be
seated but there are only 5 seats
available?

38. COMBINATION

39. COMBINATION
A way of selecting r objects out of n objects where
arrangement is not important.
The set of the different combinations formed from n
objects taken r at a time is a subset of the set of
permutation.

40. COMBINATION FORMULA
❑𝒏 𝑪𝒓=
𝒏!
(𝒏−𝒓)!𝒓!

41. EXAMPLE #1
In how many ways can 5 fruits be selected from the 7
available fruits in making fruit salad?
7C5 = =
n = 7 r =
5

42. 7C5 =
7C5 = 21 ways

43. EXAMPLE #2
In how many ways can 3 pizza toppings be selected from
the 10 toppings suggested in the menu?
10C3 = =
n = 10
r = 3

44. 10C3 =
10C3 = 120 ways

45. EXAMPLE #3
In how many ways can 7 students be selected among 12
students trying out for the school basketball varsity?
12C7 = =
n = 12
r = 7

46. 12C7 =
12C7 = 792 ways

47. EXAMPLE #4
Suppose you are given an ordinary deck of playing cards. In
how many ways can 5 cards be selected.
52C5 = =
n = 52
r = 5

48. 52C5 =
52C5 = 2, 598, 960

49. SOLVING
PERMUTATION &
COMBINATION

50. PROBLEM #2
In how many ways can 9 students
out of 14 students be chosen to be
a part of the school glee club?

51. PROBLEM #3
In the class of 30 students, 4 of them are to
be announced as the class valedictorian,
salutatorian, first honor, and second honor. If
all of them are qualified to the said titles, in
how many ways can 4 students be selected?

52. COMBINATION

53. EXAMPLE #1
A math teacher is thinking of forming a
committee that will oversee the celebration. He
would like to select two boys and three girls
from his class of 12 boys and 14 girls. How
many combinations of two boys and 3 girls are
possible?

54. EXAMPLE #2
Three colored paper strips are drawn at random from a container
containing three red and five yellow strips.
A. How many combinations of three colored paper strips can be
drawn?
B. How many combinations of three red paper strips can be drawn?
C. How many combination of two yellow paper strips and a red paper
strips can be drawn.