4.1 fcp and permutations
DESCRIPTION
TRANSCRIPT
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?