Permutationsfancy word isn’t it…

Making Groups• I need 4 volunteers

• We are going to make groups of 3 people

• Let’s make those groups

• What situations make the order important?

• How did you organize the information?

• Come up to the board and write how you recorded your information.

Fitting objects into places• You run a tanker

company

• You have n items and r places to put them

• Determine the total number of ways that you can place these items into those places

• Create a chart like the following:

1 2 3 4 5

1

2

3

4

5

n = number of objects

r =

num

ber

of s

pace

s 1 2

62 6 12 20

3 4 5

24 60

24

120

120

Does anyone see any patterns?

Think about our tree diagrams• Let’s look at the 4 pieces

(r) and 3 places (n)

• How many branches would you create for your first space to have something placed into it?

• How many branches would come off of each initial branch?

• Why is it less

each time?

A

B C DA

First Space

Second Space

Third Space

A

B

C

D

3 spaces, 7 objects• How many different arrangements could

we make from this?7 x 6 x 5

210

Since we have 7 possibilities for the first spot, 6 in the following spot and 5 in the final spot

You could also draw a tree – 7 branches for the first area, 6 off of each of those, 5 off of each of the 6…

This is called the counting principle

Factorial• So if we had 7 spaces and 7 objects, how

many arrangements would we have possible?7 x 6 x 5 x 4 x 3 x 2 x 1

We can write this as 7!

This reads: “7 factorial”

We can do this operation on our calculators

MATH PRB 4

Alphabet soup• We have a can of alphabet soup and we have one of

each letter. You are going to make a word with ALL 13 letters in the soup. (They don’t have 26 letters)

• How many different words would you have?

13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1

Which is the same as

13!

Using a calculator:

6227020800 different options

What if we don’t use all of 13!• If we had 13 letters but only wanted to make

words with 9 letters.• What operation would I do?

13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5259459200

• We have a shorter way of doing this operation

)!913(

!13

1234

12345678910111213

!4

!13

=13 x 12 x 11 x 10 x 9 x 8 x 7 x 6 x 5

Notation for Permutations• We have a formal way of writing this:• For n pieces and r spaces,

nPr =

• So for our case it would be:

13P9 =

)!(

!

rn

n

)!913(

!13

“The number of permutations of n things chosen r at a time”

Practice• Solve for the following. Write it out using

factorials then all of the steps to be multiplied, for example:

•10P3 =

• 7P2

• 5P4

• 12P5

)!310(

!10

!7

!10 = 10 x 9 x 8

= 42

= 120

= 95040

Finding Probabilities• Here is where it gets fun…I have one of each of

the letters of Alphabet Soup in my bowl • What is the probability of spelling BAT from my

bowl if I randomly select 3 letters?• So just like always, to find a probability you find

the following:

• So we have one outcome of the total number of outcomes, 1716

• P(BAT) = 1/1716

Number of desired outcomes

Total number of outcomes

License Plates• How many different

license plates can the state of Colorado produce if the first three slots are numbers, and the last three slots are letters?

• What is the probability of getting 000 AAA?

• What is the probability of getting a plate that does not repeat?

• What is the probability of getting a James Bond (007) license plate? – saw one the other day

The Band• There is a group performing

tonight and the 7 musicians performing have their names randomly printed on the program.

• What is the probability that the list is in alphabetical order?

• If we only have four chairs for the performers to sit in, how many different ways can they seat themselves?

Combination Locks• You are a lock manufacturer

and are designing locks. • How many different

combinations are available for a lock with 3 different numbers if the numbers do not repeat?

• With 4 different numbers?• How many more times “safe” is

a 4 number lock than a 3 number lock?

Homework and Reading• Read page 303 thru 304

• Read the strategy used on page 305

• p306-7 #4, 12a,b, 16-20, 24-26