Example 1B: Using the Fundamental Counting Principle

A password for a site consists of 4 digits followed by 2 letters. The letters A and Z are not used, and each digit or letter many be used more than once. How many unique passwords are possible?

digit digit digit digit letter letter

10 10 10 10 24 24 = 5,760,000

There are 5,760,000 possible passwords.

Check It Out! Example 1a

A “make-your-own-adventure” story lets you choose 6 starting points, gives 4 plot choices, and then has 5 possible endings. How many adventures are there?

A password is 4 letters followed by 1 digit. Uppercase letters (A) and lowercase letters (a) may be used and are considered different. How many passwords are possible?

Since both upper and lower case letters can be used, there are 52 possible letter choices.

letter letter letter letter number

52 52 52 52 10 = 73,116,160

There are 73,116,160 possible passwords.

A permutationis a selection of a group of objects in which order is important.

A third item C can be first, second, or third for each order

above.

1 permutation

2 · 1 permutations

3 · 2 · 1

permutations

You can see that the number of permutations of 3 items is 3 · 2 · 1. You can extend this to permutations of n items, which is n · (n – 1) · (n – 2) · (n – 3) · ... · 1. This expression is called n factorial, and is written as n!.

Sometimes you may not want to order an entire set of items. Suppose that you want to select and order 3 people from a group of 7. One way to find possible

permutations is to use the Fundamental Counting Principle.

First Person

Second Person

Third Person

There are 7 people. You are choosing 3 of them in order.

7 choices

6 choices

5 choices

=

210 permutations

arrangements of 4 4! 4 · 3 · 2 · 1

Another way to find the possible permutations is to use factorials. You can divide the total number of arrangements by the number of arrangements that are not used. In the previous slide, there are 7 total people and 4 whose arrangements do not matter.

How many ways can a student government select a president, vice president, secretary, and treasurer from a group of 6 people?

This is the equivalent of selecting and arranging 4 items from 6.

= 6 • 5 • 4 • 3 = 360

Divide out common factors.

There are 360 ways to select the 4 people.

Substitute 6 for n and 4 for r in

Example 2B: Finding Permutations

How many ways can a stylist arrange 5 of 8 vases from left to right in a store display?

Divide out common factors.

= 8 • 7 • 6 • 5 • 4

= 6720

There are 6720 ways that the vases can be arranged.

Check It Out! Example 2a

Awards are given out at a costume party. How many ways can “most creative,” “silliest,” and “best” costume be awarded to 8 contestants if no one gets more than one award?

= 8 • 7 • 6

= 336

There are 336 ways to arrange the awards.

Check It Out! Example 2b

How many ways can a 2-digit number be formed by using only the digits 5–9 and by each digit being used only once?

= 5 • 4

= 20

There are 20 ways for the numbers to be formed.

A combinationis a grouping of items in which order does not matter. There are generally fewer ways to select items when order does not matter. For example, there are 6 ways to order 3 items, but they are all the same combination:

6 permutations {ABC, ACB, BAC, BCA, CAB, CBA}

1 combination {ABC}

To find the number of combinations, the formula for permutations can be modified.

Because order does not matter, divide the number of permutations by the number of ways to arrange the selected items.

When deciding whether to use permutations or combinations, first decide whether order is important. Use a permutation if order matters and a combination if order does not matter.

You can find permutations and combinations by

using nPr and nCr, respectively, on scientific and graphing calculators.