12.1 The Fundamental Counting Principle and Permutations

How Many Ways? You want to make a sandwich for lunch. You have 4 types of meat (Ham, Turkey,

Roast Beef, Salami) and 3 types of bread (White, Wheat, Rye) to choose from.

How many different sandwiches can you make?

(Draw a tree diagram!)

Getting Dressed

You look in the closet and realize you have: only 2 clean shirts (blue and green) 4 pairs of shorts (orange, yellow, red, and

gray) 3 pairs of shoes (black, brown, and white) How many different outfits can you make?(Draw a tree diagram!)

An Easier Way? Fundamental Counting Principle (FCP):

Multiplying the number of ways each event can occur gives the number of possible outcomes.

Example: At a restaurant, you have a choice of 8

entrees, 2 salads, 12 drinks, and 6 desserts. How many different meals can you choose?

(must include one choice from each group)

Example:

A criminal identification kit contains 195 hairlines, 99 eyes, 89 noses, 105 mouths, and 74 chins.

How many different faces can be made?

A witness knows which hairline and eyes the suspect had. How many different faces can be made knowing this information?

Your Turn!

A high school has 273 freshmen, 291 sophomores, 252 juniors, and 237 seniors.

How many different ways can a committee be formed that includes 1 person from each grade?

F.C.P. with Repetition

A standard New York license plate has 3 letters followed by 3 digits.

If digits and letters can be repeated, how many possibilities are there?

If digits and letters can’t be repeated, how many?

Example:

How many different 7 digit phone numbers are possible if the first digit cannot be 1 or 0?

Your Turn!

A multiple choice test has 10 questions with 4 answer choices each. How many different ways could you complete the test?

Putting Things in Order

How many different ways can you arrange the letters A, B, and C?

Make a list.

Now, use the FCP.

An ordering of objects is called a permutation of the objects.

Finding Permutations

The # of permutations (orderings) of n distinct objects is n!

n! is read “n factorial” Factorial means:

n ∙ (n – 1) ∙ (n – 2) ∙ … ∙ 3 ∙ 2 ∙ 1 Examples: 5! 8!

Examples:

You have homework from all 6 of your classes for the weekend. In how many different orders can you complete your assignments?

There are 8 movies in theaters that you want to see. In how many different ways can you see all 8 of the movies?

Examples:

Find the number of distinct permutations of the letters in each word.

HI

JET

IOWA

GOLDFINCH

More Factorials

Evaluate:

Ordering from a Group

The number of permutations of r objects from a group of n distinct objects is denoted nPr.

nPr

Example:

You are considering 10 colleges. In how many orders can you visit 6 of them?

All 10 of them?

Example:

There are 12 books on the summer reading list. In how many orders can you read 4 of them?

All 12 of them?

Your Turn!

There are 9 players on a baseball team. In how many ways can you choose the batting order for all 9 players?

In how many ways can you choose a pitcher, catcher, and shortstop from the 9?

Permutations with Repetition

If certain objects repeat, they are not distinct anymore.

To find these permutations with repetition where n is the # of objects and q is the number of times any object repeats is:

Example:

Find the number of distinguishable permutations of the letters in:

OHIO

MISSISSIPPI

Example:

Your dog has 8 puppies, 3 are male and 5 are female. How many different birth orders are possible? (Hint: One is MMMFFFFF)

Your Turn!

A music store wants to display 3 identical keyboards, 2 identical trumpets, and 2 identical guitars. How many distinguishable displays are possible?