This document discusses permutation formulas for arranging objects in different scenarios. It covers: 1) Permutations of n objects taken n at a time is n!. 2) Permutations of n objects taken r at a time is n!/(n-r)!. 3) Distinguishable permutations when some objects are alike uses the formula n!/(p!q!r!...). 4) Circular permutations (arranging objects in a circle) is (n-1)!. Examples and solutions are provided to illustrate each type of permutation.

1. PERMUTATION
OF OBJECT

2. 1. PERMUTATION OF N OBJECTS TAKEN N AT A TIME
Formula: 𝑃(𝑛, 𝑛) = 𝑛!
Example 1: P(5,5)=?
Solution: 𝑃(5,5)
5! = 5 × 4 × 3 × 2 × 1
= 120
Example 2: In how many ways can you arrange 6 different
potted plants in a row?
Solution: 𝑃(6,6)
= 6! = 6 × 5 × 4 × 3 × 2 × 1
= 720 𝑤𝑎𝑦s
2

3. 3
2. PERMUTATION OF N
OBJECTS TAKEN R AT A TIME
Formula: 𝑃(𝑛, 𝑟) = 𝑛!
(𝑛−𝑟)! , 𝑛 ≥ 𝑟
Example 1: P (6,3) =?
Solution: 𝑃(6,3) =
6!
(6−3)!
6!
3!
6×5×4×3×2×1
3×2×1
= 120

4. Example 2:
How many ways can you arrange 3 books in the shelf
from 5 different books?
Solution: 𝑃(5,3)
5!
(5−3)!
5!
2!
5×4×3×2×1
2×1
= 60 𝑤𝑎𝑦s

5. 5
refers to the arrangements of a set of objects where
some of them are alike.
The number of distinguishable permutations of n objects
when p objects are alike, q objects are alike, r objects are
alike, and so on is given by:
Formula:
𝑃 = 𝑛!
𝑝!×𝑞!×𝑟!
3. DISTINGUISHABLE PERMUTATIONS

6. Example: How many different permutations can be made
from the word MATHEMATICS?
Solution: There are 11 letters of the word.
However, 2 M’s are alike, 2 A’s are alike, and 2 T’s are alike.
Hence, 𝑃 = 11!
2! × 2! × 2!
= 11 ∙ 10 ∙ 9 ∙ 8 ∙ 7 ∙ 6 ∙ 5 ∙ 4 ∙ 3 ∙ 2 ∙ 1
(2 ∙ 1) ∙ (2 ∙ 1) ∙ (2 ∙ 1)
= 4,989,600 𝑤𝑎𝑦s

7. 4. CIRCULAR PERMUTATION- IT IS THE
ARRANGEMENTS OF OBJECTS IN A CIRCLE.
7
- it is the arrangements of objects in a circle.
Formula: 𝑃 = (𝑛 − 1)!
Example:
How many ways can 9 students sit in a
round table?
Solution: 𝑃 = (9 − 1)! = 8! = 40,320 𝑤𝑎𝑦s

8. INDEPENDENT ACTIVITY 1: COUNT ME IN 8
Direction: Determine all the possible arrangement
of the following using the tree diagram.
1. Selecting an outfit from 2 pair of pants, 3
blouses and 2 pairs of shoes.
2. Forming a 3-digit codes using the digits 0, 1, 3
,5 if repetition is not allowed.
3. The order by which four classmates John, Paul,
Greg and Rein enter the room.

9. INDEPENDENT ASSESSMENT 1:
Directions: List all possible arrangement and
determine how many ways of the following.
1.L, O, V, E
2. X, Y, J, K, taken two at a time
3. 1, 2, 3, 4, taken three at a time
9

10. TEST YOURSELF:
1. How many arrangements are there in the
letters of the word PHONE?
2. How many 3-digit codes can be made using
the digits 0, 1,2,3,4,5,6,7,8,9?
3.How many distinguishable permutations are
possible with the letters of the word ELLIPSES?
4. In how many ways can 10 people sit in a
round table?
10

11. THANK YOU