Permutations

PermutationsPermutations

PermutationsPermutations

ObjectivesObjectives::

(1) Students will be able to use (1) Students will be able to use permutations to find all possible permutations to find all possible arrangements involving a limited arrangements involving a limited number of choices.number of choices.

Essential QuestionsEssential Questions::

(1) What are permutations and (1) What are permutations and how can we find them?how can we find them?

PermutationsPermutations

What is a PermutationWhat is a Permutation??

- Have you ever been in an ice cream - Have you ever been in an ice cream shop and wondered about all the shop and wondered about all the different ways you could order three different ways you could order three different scoops of ice cream?different scoops of ice cream?

- A - A PERMUTATIONPERMUTATION is an arrangement or is an arrangement or listing in which order listing in which order ISIS important. important.

PermutationsPermutations

Real World Example:Real World Example:

Five students are finalists in the school Five students are finalists in the school spelling bee. How many ways can spelling bee. How many ways can they finish first, second, and third?they finish first, second, and third?

PermutationsPermutations

Real World Example:Real World Example:

Five students are finalists in the school Five students are finalists in the school spelling bee. How many ways can spelling bee. How many ways can they finish first, second, and third?they finish first, second, and third?

P(5,3) = 5 x 4 x 3 = 60 P(5,3) = 5 x 4 x 3 = 60 different different waysways

PermutationsPermutations

How Do I Find The Value of A How Do I Find The Value of A PermutationPermutation??

- We calculate the value of a - We calculate the value of a permutation in the following way:permutation in the following way:

P(5,3) = 5 x 4 x 3 = 60 P(5,3) = 5 x 4 x 3 = 60 different different waysways

Start with this number

Count down this many numbers

(1) (2) (3)

PermutationsPermutations

Example 1Example 1:: Permutations.Permutations.Find the value for P(5,2).Find the value for P(5,2).

PermutationsPermutations

Example 1Example 1:: Permutations.Permutations.Find the value for P(5,2).Find the value for P(5,2).

P(5,2) = 5 x 4 = P(5,2) = 5 x 4 = 2020

Start with this number

We are using this many numbers so we count down this many numbers

(1) (2)

PermutationsPermutations

Example 2Example 2:: Standing in Line.Standing in Line.

In how many different ways can Carlos, In how many different ways can Carlos, Sergio, Caleb, DeMoris, Eric, and Sergio, Caleb, DeMoris, Eric, and Brayton stand in line?Brayton stand in line?

PermutationsPermutations

Example 2Example 2:: Standing in Line.Standing in Line.

In how many different ways can Carlos, In how many different ways can Carlos, Sergio, Caleb, DeMoris, Eric, and Sergio, Caleb, DeMoris, Eric, and Brayton stand in line?Brayton stand in line?

P(6,6) = 6P(6,6) = 6 x x 55 x x 44 x x 33 x x 22 x x 1 = 720 1 = 720 different different waysways

There are 6 people to choose from

We are selecting this many people

(1) (2) (3) (4) (5) (6)

PermutationsPermutations

Example 3Example 3:: Video Games.Video Games.

If I choose three video games to play at If I choose three video games to play at Celebration Station out of ten, in how Celebration Station out of ten, in how many different orders can I play those many different orders can I play those three games?three games?

PermutationsPermutations

Example 3Example 3:: Video Games.Video Games.

If I choose three video games to play at If I choose three video games to play at Celebration Station out of ten, in how Celebration Station out of ten, in how many different orders can I play those many different orders can I play those three games?three games?

P(10,3) = 10P(10,3) = 10 x x 99 x x 88 = 720= 720 different different ordersordersWe are selecting 3 games to play

(1) (2) (3)

There are 10 games to choose from

PermutationsPermutations

Example 4Example 4:: Arrange letters in a Arrange letters in a word.word.

In how many different ways can you In how many different ways can you arrange the letters in the word arrange the letters in the word rainbow?rainbow?

PermutationsPermutations

Example 4Example 4:: Arrange letters in a Arrange letters in a word.word.

In how many different ways can you In how many different ways can you arrange the letters in the word arrange the letters in the word rainbow?rainbow?

P(7,7) = 7P(7,7) = 7 x x 66 x x 55 x x 44 x x 33 x x 22 x x 1 = 50401 = 5040 waysways

We are selecting all 7 letters

(1) (2) (3)

There are 7 different letters to arrange

(4) (5) (6) (7)

PermutationsPermutations

Guided PracticeGuided Practice:: Find the value.Find the value.

(1) P(8,3) = ?(1) P(8,3) = ?

(2) How many ways can the three (2) How many ways can the three members of the debating team be members of the debating team be arranged on the stage?arranged on the stage?

PermutationsPermutations

Guided PracticeGuided Practice:: Find the value.Find the value.

(1) P(8,3) = (1) P(8,3) = 8 x 7 x 6 = 8 x 7 x 6 = 336336

(2) How many ways can the three (2) How many ways can the three members of the debating team be members of the debating team be arranged on the stage?arranged on the stage?

P(3,3) = 3 x 2 x 1 = P(3,3) = 3 x 2 x 1 = 6 ways6 ways

PermutationsPermutations

Independent PracticeIndependent Practice:: Find the Find the value.value.

(1) P(6,4) = ?(1) P(6,4) = ?

(2) How many ways can 4 books be (2) How many ways can 4 books be arranged on a bookshelf?arranged on a bookshelf?

PermutationsPermutations

Independent PracticeIndependent Practice:: Find the Find the value.value.

(1) P(6,4) = (1) P(6,4) = 6 x 5 x 4 x 3 = 6 x 5 x 4 x 3 = 360360

(2) How many ways can 4 books be (2) How many ways can 4 books be arranged on a bookshelf?arranged on a bookshelf?

P(4,4) = 4 x 3 x 2 x 1 = P(4,4) = 4 x 3 x 2 x 1 = 24 24 waysways

PermutationsPermutations

Real World ExampleReal World Example:: Ice Cream.Ice Cream.Coldstone Creamery has a total of 31 Coldstone Creamery has a total of 31

different flavors. They are running a different flavors. They are running a special where you can get three scoops special where you can get three scoops for the price of one. How many ways can for the price of one. How many ways can you order three different flavored scoops.you order three different flavored scoops.

PermutationsPermutations

Real World ExampleReal World Example:: Ice Cream.Ice Cream.Coldstone Creamery has a total of 31 Coldstone Creamery has a total of 31

different flavors. They are running a different flavors. They are running a special where you can get three scoops special where you can get three scoops for the price of one. How many ways can for the price of one. How many ways can you order three different flavored scoops.you order three different flavored scoops.

P(31,3) = 31 x 30 x 29 = 26,970 P(31,3) = 31 x 30 x 29 = 26,970 different different waysways

Start with this number

Count down this many numbers

(1) (2) (3)

PermutationsPermutations

SummarySummary::

- Permutations involve arrangements or - Permutations involve arrangements or listings where listings where order is importantorder is important..

- We use the following notation:- We use the following notation:

P(9,4) =P(9,4) =* The symbol P(9,4) represents the number of * The symbol P(9,4) represents the number of

permutations of 9 possible things to take, and we are permutations of 9 possible things to take, and we are taking 4 of themtaking 4 of them

PermutationsPermutations

SummarySummary::

- Permutations involve arrangements or - Permutations involve arrangements or listings where order is important.listings where order is important.

- We use the following notation:- We use the following notation:

PP((99,,44) = 9 x 8 x 7 x ) = 9 x 8 x 7 x 6 =6 =

Start with this number

Count down this many numbers

Permutation

HomeworkHomework::- Core 01 - Core 01 → p.___ #___, all→ p.___ #___, all

- Core 02 - Core 02 → p.___ #___, all→ p.___ #___, all

- Core 03 - Core 03 → p.___ #___, all→ p.___ #___, all

PermutationsPermutations