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Permutations. Permutations. Objectives : (1) Students will be able to use permutations to find all possible arrangements involving a limited number of choices. Essential Questions : (1) What are permutations and how can we find them?. Permutations. What is a Permutation ? - PowerPoint PPT Presentation
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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