1. Permutations and Combinations

2. Permutations and Combinations
Warm Up
Simplify each expression.
1. 10 • 9 • 8 • 7 •           3. 
Let a b = 2a(a + b). Evaluate each expression.
4. 3 • • • • 10
4 • 3 • 2
6 • 5
7 • 6 • 5 • 4 • 3 • 2 • 1
4 • 3 • 2 • 1 *

3. Permutations and Combinations
Solutions
1. 10 • 9 • 8 • 7 • 6 = 30,240
2. =
3. = 7 • 6 • 5 = 210
4. 3 • 4 with a • b = 2a(a + b):
2(3)(3 + 4) = 2(3)(7) = 6(7) = 42
5. 2 • 7 with a • b = 2a(a + b):
2(2)(2 + 7) = 2(2)(9) = 4(9) = 36
6. 5 • 1 with a • b = 2a(a + b):
2(5)(5 + 1) = 2(5)(6) = 10(6) = 60
7. 6 • 10 with a • b = 2a(a + b):
2(6)(6 + 10) = 2(6)(16) = 12(16) = 192
4 • 3 • 2
6 • 5
7 • 6 • 5 • 4 • 3 • 2 • 1
4 • 3 • 2 • 1
/ / / 4 5
What does this have to do with permutations or combinations?

4. Permutations vs. Combinations
Both are ways to count the possibilities
The difference between them is whether order matters or not
Does the order matters?
If YES, then we are dealing with PERMUTATIONS
If NO, then we are dealing with COMBINATIONS

5. Permutations
A permutation is an arrangement of items in a particular order. You can often find the number of permutations of some set of items using the Multiplication Counting Principle, or factorial notation
n Factorial (n!)
For any positive integer n,
n! = n(n-1) * * 3 * 2 * 1
For n=0, n!=1

6. Permutations and Combinations
Example: In how many ways can 6 people line up from left to right for a group photo?
Since everybody will be in the picture, you are using all the items from the
original set. You can use the Multiplication Counting Principle or
factorial notation.
There are six ways to select the first person in line, five ways to select the
next person, and so on.
The total number of permutations is 6 • 5 • 4 • 3 • 2 • 1 = 6!.
6! = 720
The 6 people can line up in 720 different orders.

7. Let’s Try Some Find the following factorials: a. 12! b. -21! c. 5! 4!
Is there a faster way to do this one?
5! = 5*4*3*2*1
4! = 4*3*2*1
-5.1 E 19 5

8. Permutations and Combinations
The total number of permutations is 6 • 5 • 4 • 3 • 2 • 1 = 6!.
How to use your graphing calculator to solve a factorial:
Enter the number you are taking the factorial of (e.g. 6)
Press MATH.
3. Cursor over to PRB for Probability
4. Press 4 to get ! option
5. Press ENTER.
6! = 720

9. Number of Permutations
The number of permutations of n items of a set arranged r items at a time is n P r
Where
Example: 9P5 =

10. Permutations and Combinations
Your new iPad requires a four-letter password. How many 4-letter passwords can be made if no letter can be used twice?
Method 1:  Use the Multiplication Counting Principle.
26 • 25 • 24 • 23 = 358,800
Method 2:  Use the permutation formula. Since there are 26 letters
arranged 4 at a time, n = 26 and r = 4.
26P4 = = = 358,800
26!
(26 – 4)!
22!
There are 358,800 possible arrangements of 4-letter passwords
with no duplicates.

11. Let’s Try Some
Evaluate the following:
a) 3P2 b) 8P4 c) 5P3

12. Let’s Try Some
Evaluate the following:
a) 3P2 b) 8P4 c) 5P3

13. Combinations
When the order of the items does not matter, we look for a combination. The number of n items of a set arranged r items at a time is n C r
Where
Example: 8C2 =

14. Combinations Evaluate 10C4. 10C4 = = / / / / / / = = = 210 10!
4!(10 – 4)! =
10!
4! • 6!
10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 1
4 • 3 • 2 • 1 • 6 • 5 • 4 • 3 • 2 • 1 =
/ / / / / /
10 • 9 • 8 • 7
4 • 3 • 2 • 1 =
= 210

15. Let’s Try Some
Evaluate the following:
a) 7C3 b) 25C2 c) 10C5

16. Let’s Try Some
Evaluate the following:
a) 7C3 b) 25C2 c) 10C5

17. Isn’t there an easier way???
Yes. But you need to be able to do the non-calculator method and theory too.
Let’s start with the permutation 26P4
Enter the number if items in your set (e.g. 26)
Press MATH.
3. Cursor over to PRB for Probability
4. Press 2 to get nPr option
5. Enter the arranged item set r (e.g. 4).
6. Press enter.
26P4 =

18. Let’s Retry Our Problems from Earlier using the Permutation Function on the Graphing Calculator
Evaluate the following:
a) 3P2 b) 8P4 c) 5P3
Solutions:
a) 6 b) 1680 c) 60

19. Combinations Write: nCr = 12C5 =792
A disk jockey wants to select 5 songs from a new album that contains 12 songs. How many 5-song selections are possible?
Relate: 12 songs chosen 5 songs at a time
Define: Let n = total number of songs.
Let r = number of songs chosen at a time.
Write: nCr = 12C5
=792
Similar to our permutation shortcut, you can use the combination shortcut
You can choose five songs in 792 ways.

20. Combinations
A pizza menu allows you to select 4 toppings at no extra charge from a list of 9 possible toppings. In how many ways can you select 4 or fewer toppings?
You may choose 
4 toppings, 3 toppings, 2 toppings, 1 toppings, or none.
9C4 9C3 9C2 9C1 9C0
The total number of ways to pick the toppings is
= 256.
There are 256 ways to order your pizza.

21. Let’s Retry Our Problems from Earlier using the Combination Function on the Graphing Calculator
Evaluate the following:
a) 7C3 b) 25C2 c) 10C5
Solutions:
a) 35 b) 480,700 c) 252

22. In Summary
How can you decide if you have a permutation or a combination?
Permutations use the entire group and order matters
Ex: How many ways can you arrange all the students in the class into one line?
Combinations use parts of the group and the order does not matter
Ex: If you have 10 people in a group, how many one on one conversations can you have?