Permutations

Chapter 12Section 7

Your mom has an iTunes account. You know iTunes requires her to enter a 7-letter password. If her password is random, is it possible for you to guess the correct 7-letter combination so you can access her account and buy your favorite album?

Counting

You have three shirts, two pairs of pants, and three hats. How many outfits can you make

consisting of one shirt, one pair of pants, and one hat?

How can we guarantee that we count them all?

• By using a tree diagram.

• Suppose there are 2 ways to get from City A to City B.

• There are 4 ways to get from City B to City C.• There are 3 ways to get from City C to City D.

• How many different routes are there from City A to City D.

• Joe has 3 shirts and 2 ties (that all match). How many outfits can he make?

• At “Joe’s Pizza,” there are 5 vegetable toppings and 3 meat toppings. How many pizzas can Joe make with 1 vegetable and 1 meat topping?

• At the library, there are 4 of Joe’s favorite sci-fi novels, and 4 of his favorite mystery novels. He wants to read one of each. How many different sets of 1 sci-fi and 1 mystery book can he pick?

The Multiplication Counting Principle

If there are m ways to make 1 selection, and n ways to make a second selection, then there are m x n ways to make the two selections.

For example: Joe had 3 shirts and 2 ties. So, m=3

and n=2. The number of shirt-tie pairs he can make is: m x n = 3 x 2 = 6.

Does Order Matter?

• You’re making a sandwich with turkey, lettuce, an cheese.

• Your password consists of 5 letters.

Permutations: When Order Matters

• A permutation is an arrangement of objects in a particular order.

• How many ways can you arrange the letters A, B, and C without repeating?

• How many different batting orders can you have with 9 baseball players?

Try These:• A 7-digit telephone number can begin with

any digit except 0 or 1. How many possible choices are there for each of the 7 digits?– How many different 7-digit telephone numbers

are possible?

• How many ways can you seat 5 people in a row?

Permutation Notation

• The expression nPr represents the number of permutations of n objects arranged r at a time.

• nPr = (n)(n-1)(n-2)…(stop when you have r factors)

• For example: 8P3 represents 8 objects chosen 3 at a time. – 8P3 = (8)(8-1)(8-2)=(8)(7)(6)=336

Factorial

• Another way to write:n(n-1)(n-2)(n-3) … (1)

is n!, and we say “n factorial”

• For example: 4! = 4(3)(2)(1) = 24

• What is 6!

Try These:

• Calculate the following:– 5P4

– 8P5

• How many ways can you arrange 6 letters in groups of 4?

Using Your Calculator

• Type in your n.• Select MATH• Select nPr.• Type in your r.• Hit ENTER.

• Use your calculator to find: 4P4, 8P5, and 9P6

Your mom has an iTunes account. You know iTunes requires her to enter a 7-letter password. If her password is random, is it possible for you to guess the correct 7-letter combination so you can access her account and buy your favorite album?

Let’s Try This Again:

Exit Ticket

• The companies listed on the New York Stock Exchange have a 1, 2, or 3 letter code. How many companies can be listed?