6.7 – Permutations 6.7 – Permutations and Combinationsand Combinations

Permutations and Permutations and CombinationsCombinations

Simplify each expression.

1. 10 • 9 • 8 • 7 • 6 2.           3. 

Let a b = 2a(a + b). Evaluate each expression.

4. 3 • 4 5. 2 • 7 6. 5 • 1 7. 6 • 10

4 • 3 • 26 • 5

7 • 6 • 5 • 4 • 3 • 2 • 14 • 3 • 2 • 1

*

Warm Up

Permutations and Permutations and CombinationsCombinations

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 • 26 • 5

7 • 6 • 5 • 4 • 3 • 2 • 14 • 3 • 2 • 1

/ // /

/ //

45

/ // /

What does this have to do with permutations or combinations?

PermutationsPermutations A permutation is an arrangement of A permutation is an arrangement of

items in a particular order. You can often items in a particular order. You can often find the number of permutations of some find the number of permutations of some set of items using the Multiplication set of items using the Multiplication Counting Principle, or factorial notationCounting Principle, or factorial notation

n Factorial (n!)

For any positive integer n,

n! = n(n-1) * . . . * 3 * 2 * 1

For n=0, n!=1

Example: In how many ways can 6 people line up from left to

right for a group photo?

Permutations and Permutations and CombinationsCombinations

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.

Let’s Try SomeLet’s Try Some

Find the following factorials:Find the following factorials:

a.a. 12!12! b.b. -21!-21! c.c. 5!5!

4!4!

a.479001600

b.-5.1 E 19

c.5

Is there a faster way to do this one?

5! = 5*4*3*2*1

4! = 4*3*2*1

Permutations and Permutations and CombinationsCombinations

The total number of permutations is 6 • 5 • 4 • 3 • 2 • 1 = 6!.How to use your graphing calculator to solve a factorial:

1. Enter the number you are taking the factorial of (e.g. 6)

2. Press MATH. 3. Cursor over to PRB for

Probability 4. Press 4 to get ! option5. Press ENTER.

6! = 720

Number of PermutationsNumber of Permutations The number of permutations of n items of The number of permutations of n items of

a set arranged r items at a time is n P ra set arranged r items at a time is n P r

)!(

!Pr

rn

nn

Where nr 0

Example: 9P5 = 1512056789!4

!9

)!59(

!9

Your new iPad requires a four-letter password. How many 4-letter passwords can be made if no letter can be used twice?

Permutations and Permutations and CombinationsCombinations

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.

There are 358,800 possible arrangements of 4-letter passwords with no duplicates.

26P4 = = = 358,800 26! (26 – 4)!

26!22!

Let’s Try SomeLet’s Try Some

Evaluate the following:Evaluate the following:

a)a) 33PP22 b)b) 88PP44 c)c) 55PP33

Let’s Try SomeLet’s Try Some

Evaluate the following:Evaluate the following:

a)a) 33PP22 b)b) 88PP44 c)c) 55PP33

CombinationsCombinations When the order of the items does not When the order of the items does not

matter, we look for a matter, we look for a combinationcombination. The . The number of n items of a set arranged r number of n items of a set arranged r items at a time is n C ritems at a time is n C r

)!(!

!

rnr

nnCr

Where nr 0

Example: 8C2 = 287202

40320

!6!2

!8

)!28(!2

!8

CombinationsCombinationsEvaluate 10C4.

10C4 = 10! 4!(10 – 4)!

= 10! 4! • 6!

10 • 9 • 8 • 7 • 6 • 5 • 4 • 3 • 2 • 14 • 3 • 2 • 1 • 6 • 5 • 4 • 3 • 2 • 1= / / / / / /

/ / / / / /

10 • 9 • 8 • 74 • 3 • 2 • 1=

= 210

Let’s Try SomeLet’s Try Some

Evaluate the following:Evaluate the following:

a)a) 77CC33 b)b) 2525CC22 c)c) 1010CC55

Let’s Try SomeLet’s Try Some

Evaluate the following:Evaluate the following:

a)a) 77CC33 b)b) 2525CC22 c)c) 1010CC55

Isn’t there an easier way???Isn’t there an easier way???

Let’s start with the permutation 26P4

1. Enter the number if items in your set (e.g. 26)

2. Press MATH. 3. Cursor over to PRB for

Probability 4. Press 2 to get nPr option5. Enter the arranged item

set r (e.g. 4). 6. Press enter.

26P4 = 358800

Yes. But you need to be able to do the non-calculator method and theory too.

Let’s Retry Our Problems from Earlier Let’s Retry Our Problems from Earlier using the Permutation Function on the using the Permutation Function on the

Graphing CalculatorGraphing Calculator

Evaluate the following:Evaluate the following:

a)a) 33PP22 b)b) 88PP44 c)c) 55PP33

Solutions:

a) 6 b) 1680 c) 60

CombinationsCombinationsA 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 timeDefine: Let n = total number of songs.

Let r = number of songs chosen at a time.

You can choose five songs in 792 ways.

Write: nCr = 12C5

Similar to our permutation shortcut, you can use the combination shortcut

=792

CombinationsCombinationsA 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?

The total number of ways to pick the toppings is 126 + 84 + 36 + 9 + 1 = 256.

There are 256 ways to order your pizza.

You may choose 4 toppings, 3 toppings, 2 toppings, 1 toppings, or none.

9C4 9C3 9C2 9C1 9C0

Let’s Retry Our Problems from Earlier Let’s Retry Our Problems from Earlier using the Combination Function on the using the Combination Function on the

Graphing CalculatorGraphing Calculator

Evaluate the following:Evaluate the following:

a)a) 77CC33 b)b) 2525CC22 c)c) 1010CC55

Solutions:

a) 35 b) 480,700 c)252

In SummaryIn Summary How can you decide if you have a How can you decide if you have a

permutation or a combination?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?