1. Permutations
Section 12.5

2. Permutations ORDER IS IMPORTANT How many unique ways to order a group
Winning 1st and 2nd prize. Different??
1,2,3 is different from 1,3,2 and 2,1,3
How many unique ways to order a group
How many ways to arrange the students at lineup
How many possible outcomes in an election

3. Permutation Formula Permutation Formula nPr : n! / (n-r)!
Example4P3 = 4! / (4-3)!
8P5 = 8! / (8-5)!
9P3 = 9! / (9-3)!
11P2
9P9
11P0

4. Cancelling Factorials
7! / 4!
Not necessary to write (7*6*5*4*3*2*1)/ (4*3*2*1)
Only need to write to highest factorial in the denominator.
Change 7! To 7*6*5*4!
Then divide by 4!
Result 7*6*5
9! / 5!
Change 9! To 9*8*7*6*5! Then divide by 5!
9*8*7*6

5. More Examples 9P2 = 9! / (9-2)! 9! / 7! 9*8*7! / 7! 9*8
9*8*7! / 7!
9*8
4P3 = 4! / (4-3)!
(4*3*2*1) / 1!
4P4= 4! / (4-4)!
4*3*2*1 / 0!
Remember 0! = 1

6. Checking
9P2
7P3
6P6

7. Examples
How many different ways can you arrange the letters in the word HOLIDAY?
How many letters do we have?
How many letters are we arranging?
What is the formula?
What is the answer(in factorial form)?

8. Examples
A deck of cards have 52 cards. How many ways can you arrange 5 cards.
How many cards to we have?
How many cards are we arranging?
Write the formula?
Solve (in factorial form)

9. I want to select President and Vice President from this class?
How many people are in the class?
How many are being selected?
What is the formula?
What is the answer(in factorial form)?

10. Permutation Examples
There are 8 students in a play. How many ways can you arrange 6 students to come onto the stage?
You want to arrange the entire class (33 students) in a line. How many different ways to you arrange the students?

11. More examples
You want to elect a class leader and backup leader from the class. How many different can you make this selection?

12. Permutations comparisons
Using logic, choose a comparison operator(<, <=, >, >=, =)
Same sample size, different number being arranged
6P P3
9P P2
Larger sample size, but same number chosen
8P P2
Arrange 10 people in a line ___ arrange 9 people in a line

13. Permutations with duplications
How many ways to arrange the letters in the word BANANNA?
How many letters?
How many are being arranged?
7P7
Is it a different arrangement between the 2 letter A’s and the 3 letter N?

14. Removing duplicates BANANNA
How many ways can we arrange the 2 letters A’s?
How many ways to arrange the 2 letter A’s?
?P?
How many ways to arrange the 3 letter N’s?

15. Final Solution
Number of unique ways to arrange the letters in the word BANANNA
7P7 = 7*6*5*4*3*2*1
Removing duplicates A’s
2P2
Removing duplicates N’s
3P3
Final solution: 7∗6∗5∗4∗3∗2∗1 (3∗2∗1)(2∗1)