Fundamental Counting Principle can be used determine the number of possible outcomes when there are two or more characteristics .

Fundamental Counting Principle states that if an event has m possible outcomes and another independent event has n possible outcomes, then there are m* n possible outcomes for the two events together.

Fundamental Counting Principle

Lets start with a simple example.

A student is to roll a die and flip a coin. How many possible outcomes will there be?

Fundamental Counting Principle

For a college interview, Robert has to choose what to wear from the following: 4 slacks, 3 shirts, 2 shoes and 5 ties. How many possible outfits does he have to choose from?

Example 1A: Using the Fundamental Counting Principle

To make a yogurt parfait, you choose one flavor of yogurt, one fruit topping, and one nut topping. How many parfait choices are there?

Yogurt Parfait

(choose 1 of each)

Flavor

Plain

Vanilla

Fruit

Peaches

Strawberries

Bananas

Raspberries

Blueberries

Nuts

Almonds

Peanuts

Walnuts

Example 1B: Using the Fundamental Counting Principle

A password for a site consists of 4 digits followed by 2 letters. Each digit or letter ma be used more than once. How many unique passwords are possible?

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

A Permutation is an arrangement of items in a particular order.

Notice, ORDER MATTERS!

To find the number of Permutations of n items, we can use the Fundamental Counting Principle or factorial notation.

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!.

A FACTORIAL is a counting method that uses consecutive whole numbers as factors.

The factorial symbol is !

Examples 5! = 5x4x3x2x1

=

7! = 7x6x5x4x3x2x1

=

Factorial !

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

7 choices

6 choices

5 choices

=

arrangements of 4 4!

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?

Example 2B: Finding Permutations

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

Permutations

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?

Permutations

A combination lock will open when the right choice of three numbers (from 1 to 30, inclusive) is selected. How many different lock combinations are possible assuming no number is repeated?

Practice:

Combinations

A Combination is an arrangement of items in which order does not matter.

ORDER DOES NOT MATTER!

Since the order does not matter in combinations, there are fewer combinations than permutations.

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

1 combination {ABC}

***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.***

Combinations

To play a particular card game, each player is dealt five cards from a standard deck of 52 cards. How many different hands are possible?

Practice:

Combinations

A student must answer 3 out of 5 essay questions on a test. In how many different ways can the student select the questions?