• You need to get a new cell phone. You jumped into the pool with your old one. There are 12 different models. Those models come in two different colors. And those phones come with four different service plans. How many different arrangements can you have?

• You are taking a multiple-choice test that has twelve questions. Each of the questions has four choices, with one correct choice per question. If you select one of these options per question and leave

nothing blank, in

how many ways

can you answer the questions?

Thinking Mathematically

I can use the Fundamental Counting Principal to count Permutations.

I can evaluate factorial expressions.

I can use the permutations formula.

PermutationsA permutation is an ordered

arrangement of items that occurs when

• No item is used more than once.

• The order of arrangement makes a difference.

Example Counting Permutations

Based on their long-standing contribution to rock music, you decide that the Rolling Stones should be the last group to perform at the four-group Offspring, Pink Floyd, Sublime, Rolling Stones concert. Given this decision, in how many ways can you put together the concert?

Solution

We use the Fundamental Counting Principle to find the number of ways you can put together the concert. Multiply the choices:

3 2 1 1 = 6

Thus, there are six different ways to arrange the concert if the Rolling Stones are the final group to perform.

Example Counting Permutations

You need to arrange seven of your favorite books along a small shelf. How many different ways can you arrange the books, assuming that the order of the books makes a difference to you?

Solution

You may choose any of the seven books for the first position on the shelf. This leaves six choices for second position. After the first two positions are filled, there are five books to choose from for the third position, four choices left for the fourth position, three choices left for the fifth position, then two choices for the sixth position, and only one choice left for the last position.

7 6 5 4 3 2 1 = 5040There are 5040 different possible permutations.

• You are hosting a party and will have time to play 5 CDs. You have 35 CDs from which to choose. In how many ways can you play the music for the party if you don’t repeat any of the CDs?

Factorial Notation

If n is a positive integer, the notation n! is the product of all positive integers from n down through 1.

n! = n(n-1)(n-2)…(3)(2)(1)

0!, by definition is 1.

0!=1

• Evaluate each expression

13!

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

8*7*6*5*4*3*2*1

13*12*11*10*9*(8*7*6*5*4*3*2*1)

(8*7*6*5*4*3*2*1)

13*12*11*10*9

1 154,440

Permutations of n Things Taken r at a Time

The number of permutations possible if r items are taken from n items:

• You and 19 of your friends have decided to form an Internet marketing consulting firm. The group needs to choose three officers – a CEO, an operating manager, and a treasurer. In how many ways can those offices be filled?

!

( )!n r

nP

n r

20 3

20!

(20 3)!P

20 3

20!

17!P

20 3

20*19*18*17!

17!P

20 3

20*19*18

1P

20 3 6,840P

Kelsey Conatser• The teachers have to choose a

committee of 5 students to test the new curriculum. There are 750 students eligible for the committee. How many different committee possibilities do they have?

Jennifer DeBruce• You and 16 of your coworkers have

decided to quit and start a bakery. You need 8 positions, a treasurer, head baker, baker’s apprentice, head cake froster, 2 froster apprentices, cashier, and a person to clean. How many different arrangements can be made?

Jeffrey Rains

• You and 7 friends are going 4-wheeler riding. There are four 4-wheelers. How many different combinations

are there for

riding?

Abigail Jennings

• 20 People are going hiking. They need 5 leaders to carry food. How many different arrangements of people carrying food can be made?

Kelli Brandon

• You have ten pairs of shoes in your closet. You are going on a trip where you can only pack five pairs. How many different arrangements of shoes can you pack?

Galen Collins

• There are 29 other people in your class. You and your classmates have to pick a President, Vice President, Secretary, and Sergeant of Arms. How many

options do you have?

Savannah Sells• You and five of your friends are going

bowling. The bowling alley has 15 different types of balls to choose from. How many

different outcomes

could you have?

Cory Smith & Dustin Garrett

• There are 25 people wanting to run for six offices, President, VP, Treasurer, Secretary, Reporter, Sentinel. How many ways can these offices be filled?

Johnrick Bishop & Patrick Thurman

• You and eleven of your friends are in a competition to steal 6 cars. In how many different possible arrangements can the cars be stolen?

Jesse Boles & Brandon Cravens

• You are putting together a small army of 50 people. You need 5 leaders. How many different possible arrangements of leaders can you have?

Permutations of Duplicate Items

The number of permutations of n items, where p items are identical, q items are identical, r items are identical, and so on, is given by

!...!!

!

rqp

n

• In how many distinct ways can the letters of the word MISSISSIPPI be arranged?

!...!!

!

rqp

n

11!

4!4!2!

11*10*9*8*7*6*5*4*3*2*1

4*3*2*1*4*3*2*1*2*1

11*10*9*8*7*6*5

4*3*2*1*2*1

34,650

Thinking Mathematically

Permutations