Unit 4 – Probability and Data Analysis

4.1 The FCP and Permutations

How Many Ways?You want to make a sandwich: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!)

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:A criminal identification kit contains

195 hairlines, 99 eyes, 89 noses, 105 mouths, and 74 chins.

How many different faces can be made?

You Try!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 lettersfollowed by 3 digits.

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

How many if digits and letters can’t be repeated?

Example:How many different 7 digit phone

numbers are possible if the first digit cannot be 1 or 0?

You Try!A multiple choice test has 10

questions with 4 answer choices each. How many different ways could you complete the test?

Putting Things in OrderHow many different ways can you

arrange the letters A, B, and C?Make a list.

Now, use the FCP.

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 ∙ 1Examples:5! 8!

Examples:In how many different orders can

you complete 6 homework assignments?

Examples:Find the number of distinct

permutations of the letters in each word.

IOWA

FLORIDA

You Try!How many ways can you line up 9

people for a picture?

Ordering from a GroupThe 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?