This document provides an overview of permutations including definitions, examples of calculating permutations using factorials and the nPr notation, and applications of permutations such as arranging objects, selecting a subset of objects, and circular arrangements. It discusses permutations of n objects, distinguishable permutations, and permutations of n objects taken r at a time. Examples are provided to illustrate calculating permutations in different scenarios as well as applications and non-applications of permutations.

1. Teacher III
MATHEMATICS 10
Weeks 1-2 of Quarter 3

2. Note
Please prepare your scientific
calculator as it will be used in
our online discussion later.

3. PERMUTATION
An arrangement of objects in a definite
order or the ordered arrangement of
distinguishable objects without allowing
repetitions among the objects.

4. Preliminary Task
Using your scientific calculator, find the
value of the following:
1. 4! 24
2. 3!5! 720
3.
7!
5!
42
4.
8!
2!4!
840
5.
7!
3!4!
∙
5!
2!
2 100

5. What is meant by 𝑛! (𝑛 factorial)
𝑛! = 𝑛(𝑛 − 1)(𝑛 − 2)(𝑛 − 3) …
Example:
4! = 4 3 2 1
4! = 24

6. Types of Permutation
1. Permutation of 𝑛 objects
2. Distinguishable Permutation
3. Permutation of 𝑛 objects taken 𝑟 at a time
4. Circular Permutation

7. Permutation of 𝑛 objects
𝒏𝑷𝒏 = 𝒏!
(The permutation of 𝑛 objects is
equal to 𝑛 factorial)

8. How many arrangements are there?
1. Arranging different 3 portraits on a wall
𝟑! = 𝟔

9. How many arrangements are there?
2. Arranging 4 persons in a row for a picture taking
𝟒! = 𝟐𝟒
3. Arranging 5 different figurines in a shelf
𝟓! = 𝟏𝟐𝟎
4. Arranging 6 different potted plants in a row
𝟔! = 𝟕𝟐𝟎
5. Arranging the digits of the number 123456789
𝟗! = 𝟑𝟔𝟐 𝟖𝟖𝟎
6. Arranging the letters of the word CHAIRWOMEN
𝟏𝟎! = 𝟑 𝟔𝟐𝟖 𝟖𝟎𝟎

10. Distinguishable Permutation
𝑷 =
𝒏!
𝒑!𝒒!𝒓!
There are repeated (or identical) objects in
the set.

11. How many arrangements are there?
1. Arranging the digits in the number 09778210229
𝑃 =
11!
2! 2! 2! 3!
= 831 600
2. Drawing one by one and arranging in a row 4
identical blue, 5 identical yellow, and 3 identical
red balls in a bag
𝑃 =
12!
3! 4! 5!
= 27 720

12. How many arrangements are there?
3. Arranging the letters in the word LOLLIPOP
𝑃 =
8!
3! 2! 2!
= 1680
4. Arranging these canned goods
𝑃 =
10!
3! 4!
= 25 200

13. Permutation of 𝑛 objects taken 𝑟
𝒏𝑷𝒓 =
𝒏!
𝒏−𝒓 !
Permutation of 𝑛 taken 𝑟 at a time where
𝑛 ≥ 𝑟

14. How many arrangements are there?
1. Choosing 3 posters to hang on a wall from 5 posters
you are keeping
𝒏𝑷𝒓 =
𝒏!
𝒏−𝒓 !
𝟓𝑷𝟑 =
𝟓!
𝟓−𝟑 !
𝟓𝑷𝟑 =
𝟓!
𝟐!
𝟓𝑷𝟑 = 𝟔𝟎
Notation: 5P3 / P(5,3) Calculator: 5 shift 𝒏𝑷𝒓 3 =

15. How many arrangements are there?
2. Taking two-letter word, without repetition of letters,
from the letters of the word COVID
Examples: CO, OC, VD, …
𝒏𝑷𝒓 =
𝒏!
𝒏−𝒓 !
𝟓𝑷𝟐 =
𝟓!
𝟓−𝟐 !
𝟓𝑷𝟐 =
𝟓!
𝟑!
𝟓𝑷𝟐 = 𝟐𝟎
Notation: 5P2 / P(5,2) Calculator: 5 shift 𝒏𝑷𝒓 2 =

16. How many arrangements are there?
3. Taking four-digit numbers, without repetition of
digits, from the number 345678
Examples: 3456, 6534, 6745, …
𝒏𝑷𝒓 =
𝒏!
𝒏−𝒓 !
𝟔𝑷𝟒 =
𝟔!
𝟔−𝟒 !
𝟔𝑷𝟒 =
𝟔!
𝟐!
𝟔𝑷𝟒 = 𝟑𝟔𝟎
Notation: 6P4 / P(6,4) Calculator: 6 shift 𝒏𝑷𝒓 4 =

17. How many arrangements are there?
4. Pirena, Amihan, Alena, and Danaya competing for 1st,
2nd, and 3rd places in spoken poetry
Example: Danaya–1st, Alena–2nd, Amihan–3rd
𝒏𝑷𝒓 =
𝒏!
𝒏−𝒓 !
𝟒𝑷𝟑 =
𝟒!
𝟒−𝟑 !
𝟒𝑷𝟑 =
𝟒!
𝟏!
𝟒𝑷𝟑 = 𝟐𝟒
Notation: 4P3 / P(4,3) Calculator: 4 shift 𝒏𝑷𝒓 3 =

18. How many arrangements are there?
5. Electing Chairperson, Vice Chairperson, Secretary,
Treasurer, Auditor, PRO, and Peace Officer from a
group of 20 people
𝒏𝑷𝒓 =
𝒏!
𝒏−𝒓 !
𝟐𝟎𝑷𝟕 =
𝟐𝟎!
𝟐𝟎−𝟕 !
𝟐𝟎𝑷𝟕 =
𝟐𝟎!
𝟏𝟑!
𝟐𝟎𝑷𝟕 = 𝟑𝟗𝟎 𝟕𝟎𝟎 𝟖𝟎𝟎
Notation: 20P7 / P(20,7) Calculator: 20 shift 𝒏𝑷𝒓 7 =

19. Circular Permutation
𝑷 = (𝒏 − 𝟏)!

20. How many arrangements are there?
How many ways can 5 people sit around a circular
table?
𝑃 = (5 − 1)!
𝑃 = 4!
𝑃 = 24

22. Other Problems Involving Permutations:
1. There are 3 different History books, 4 different
English books, and 8 different Science books. In how
many ways can the books be arranged if books of the
same subjects must be placed together?
𝑃 = 3! 4! 8! 3!
𝑃 = 34 836 480

23. Other Problems Involving Permutations:
2. Three couples want to have their pictures taken. In how
many ways can they arrange themselves in a row if
couples must stay together?
𝑃 = 3! 2!
𝑃 = 12

24. Other Problems Involving Permutations:
3. In how many ways can 8 people arrange themselves in
a row if 3 of them insist to stay together?
𝑃 = 6! 3!
𝑃 = 4 320

25. Other Problems Involving Permutations:
4. In how many ways can the letters of the word
ALGORITHM be arranged if the vowel letters are
placed together?
Example: AOILGRTHM
𝑃 = 7! 3!
𝑃 = 30 240

26. Other Problems Involving Permutations:
5. In how many ways can the letters of the word
TIKTOKERIST be arranged if the consonant letters
are placed together?
Example: TKTKRSTIOEI
𝑃 =
5!
2!
∙
7!
2! 3!
𝑃 = 25 200

27. Other Problems Involving Permutations:
6. In how many ways can 7 people be seated around a
circular table if 3 of them insist on sitting beside each
other?
𝑃 = 5 − 1 ! ∙ 3!
𝑃 = 144

29. Applications of Permutations
1. Using passwords
022988, 228890
2. Using PIN of ATM cards
1357, 3715
3. Winning in a contest
Erika – 1st, Carrie – 2nd, Agatha – 3rd
Agatha – 1st, Erika – 2nd, Carrie – 3rd

30. Applications of Permutations
4. Electing officers in an organization
Gerald – President, Julia – VP, Bea – Secretary
Bea – President, Gerald – VP, Julia – Secretary
5. Assigning of telephone/mobile numbers
09778210229, 09228210779
6. Assigning plate numbers of vehicles
ABY 8512, BAY 1258

32. NOT Applications of Permutations
1. Selecting numbers in a lottery
6-12-25-32-34-41, 34-25-41-6-32-12
2. Selecting fruits for salad
apple, pineapple, grapes, papaya, pears
papaya, pears, apple, pineapple, grapes
3. Choosing members of a committee
Marissa, Ellice, Gabriel, Lucinda
Lucinda, Gabriel, Ellice, Marissa

33. NOT Applications of Permutations
4. Using points on a plane to form a
polygon (no three points are collinear)
.
.
.
.
.
. .
B
A
C
D
E
G
F

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