The document outlines the concept of linear permutations, describing rules for arranging distinct objects either all at once or a specific number at a time. It provides various examples to demonstrate how to calculate permutations using the formulas npn = n! for all objects and nPr = n!/(n-r)! for a subset of objects. Additionally, the document presents practical problems involving permutations related to books, committee selections, and rankings in contests.

1. LINEAR
PERMUTATI

2. LINEAR PERMUTATION
- this is an ordered arrangements of objects
in a line.
Rule no. 1: taken all at once
The number of permutations of ‘n’
distinct objects taken all at once is
denoted by nPn and defined as nPn =

3. Rule no. 1
Examples:
• In how many ways can 3 different books
be arranged in a shelf?
• How many possible arrangements are
there in the word MEAT?
• In how many possible arrangement can
Anna, Jane and Mary sit on a bench?

4. Rule no. 2: taken r at a time
The number of permutations of
‘n’ distinct objects taken r at a
time is defined as nPr and defined
as nPr =
𝒏!
𝒏−𝒓 !
LINEAR PERMUTATION

5. Rule no. 2
Examples:
• In how many ways can 3 different books
be arranged in a shelf if in there are only
two spaces left?
• In how many possible arrangement can 5
girls be seated in a 3-seater bench?
• In how many ways can 8 sangguniang
bayan members be selected from 15

6. Rule no. 2: Taken r at a time
The number of permutations of ‘n’ distinct
objects taken r at a time is defined as nPn and
defined as nPr =
𝒏!
Rule no. 1: Taken all at once
The number of permutations of ‘n’ distinct
objects taken all at once is denoted by nPn
and defined as nPn = 𝒏!
LINEAR PERMUTATION

7. Example 1. In how many ways can 3 different books be
arranged in a shelf?
Step 1. Analyze the problem and identify what rule to use.
Since we are going to arrange all the data at once we are going to use
rule no. 1. nPn = 𝒏!
Step 2. Identify the given in the problem.
In the problem, we have 3 books to arrange therefore our
n=3.
Step 3. Solve the problem

8. Step 3. Solve the problem
Solution:
n=3
nPn = 𝒏!
3P3 = 3!
3P3 = 3●2 ●1
3P3 = 6 ways Therefore, there are 6 different
ways to arrange the 3 books.

9. Example 1. In how many ways can 3 different books be
arranged in a shelf if there are only two spaces left?
Step 1. Analyze the problem and identify what rule to use.
In the problem we are asked to arrange 3 different books.
However, we can only arrange two different books at a time. There
we need to use rule no. 2.
nPr =
𝒏!
𝒏−𝒓 !
Step 2. Identify the given in the problem.
In the problem, we have 3 books to arrange but we can only
take 2 books at a time therefore our n=3 and r=2.

10. Step 3. Solve the problem
Solution:
n = 3 and r = 2
Therefore, there are 6 ways to
arrange the 3 different books if there
are only two spaces left .
nPr =
𝒏!
𝒏−𝒓 !
3P2 =
𝟑!
𝟑−𝟐 !
3P2 =
𝟑!
𝟏 !
3P2 =
𝟑∙𝟐∙𝟏
𝟏
3P2 =
𝟔
𝟏
or 6 ways

13. How many ways can the
letters A, B, and C be
arranged?
Answer: Rule no. 1: Taken all at
once

14. A committee of 5 members needs
to be formed from a group of 10
people. In how many ways can
the committee be selected?
Answer: Rule no. 2: Taken r at a
time

15. In how many ways can a
president, a treasurer and a
secretary be chosen from among
8 candidates?
Answer: Rule no. 2: Taken r at a
time

16. In how many ways can 4 different
books be arranged on a shelf?
Answer: Rule no. 1: Taken all at
once

17. In a singing competition with 10
contestants, in how many ways
can the organizer arrange the
first three singers?
Answer: Rule no. 2: Taken r at a
time

18. Solve the following problems involving
permutations.
1. How many ways can the letters A, B, and C be arranged?
2. A committee of 5 members needs to be formed from a group of
10 people. In how many ways can the committee be selected?
3. In how many ways can a president, a treasurer and a secretary
be chosen from among 8 candidates?
4. In how many ways can 4 different books be arranged on a
shelf?

19. Siocon National Science High School held a
singing contest during the Values Month
Celebration. Each grade level has one
representative.
a) In how many ways can the participants be
ranked from 1st for 6th place?
b) If only the top 3 finalists can receive
awards, in how many different ways can 1st,
2nd and 3rd be awarded be awarded?

20. c) A license plate begins with three letters. If
the possible letters are A, B, C, D and E, how
many different permutations of these letters
can be made if no letter is used more than
once?
d) In how many ways can a president, a
treasurer and a secretary be chosen from
among 7 candidates?

21. 8 cyclists participated in the Regional
Qualifying Meet.
a) In how many ways can the participants
be ranked from 1st for 8th place?
b) If only the top 3 finalists can move to
the Palarong Pambansa, in how many
different ways can 1st, 2nd and 3rd ranks
be awarded?