Transcript 12_2 Permutations and Combinations_ I

Exploration - Permutation
 Suppose that a manager of a softball team is filling out
her team’s lineup card before the game, the order in
which the names are filled out is important as it
determines the order which the players will bat.
 Suppose that she has 7 possible players in mind for the
top 4 spots in the lineup. How many different ways can
the manager fill in the line up?
Exploration - Permutation
 Suppose that a manager of a softball team is filling out
her team’s lineup card before the game, the order in
which the names are filled out is important as it
determines the order which the players will bat.
 Suppose that she has 7 possible players in mind for the
top 4 spots in the lineup. How many different ways can
the manager fill in the line up?
________
First
_______
Second
_______
Third
_______
Fourth
_____
Total
Exploration - Permutation
 Suppose that a manager of a softball team is filling out
her team’s lineup card before the game, the order in
which the names are filled out is important as it
determines the order which the players will bat.
 Suppose that she has 7 possible players in mind for the
top 4 spots in the lineup. How many different ways can
the manager fill in the line up?
7
________
First
6
_______
Second
5
_______
Third
4
_______
Fourth
840
_____
Total
Permutations – Definition
 When a group of objects or people are arranged in a
certain order, the arrangement is called a
. In a permutation, the order of the
objects is very important.
 The number of ways to arrange 7 people or objects
taken 4 at a time is written 𝑃(7,4).
𝑃(𝑛, 𝑟)
 Definition
 𝑃 𝑛, 𝑟 =
𝑛!
𝑛−𝑟 !
 With 𝑛! = 𝑛 ∗ 𝑛 − 1 ∗ 𝑛 − 2 ∗ ⋯ ∗ 3 ∗ 2 ∗ 1
𝑃(𝑛, 𝑟)
 Definition
 𝑃 𝑛, 𝑟 =
𝑛!
𝑛−𝑟 !
 With 𝑛! = 𝑛 ∗ 𝑛 − 1 ∗ 𝑛 − 2 ∗ ⋯ ∗ 3 ∗ 2 ∗ 1
 Example: 5! = 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 = 120
𝑃(𝑛, 𝑟)
 Definition
 𝑃 𝑛, 𝑟 =
𝑛!
𝑛−𝑟 !
 With 𝑛! = 𝑛 ∗ 𝑛 − 1 ∗ 𝑛 − 2 ∗ ⋯ ∗ 3 ∗ 2 ∗ 1
 Example: 5! = 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 = 120
 Example: 7! =
𝑃(𝑛, 𝑟)
 Definition
 𝑃 𝑛, 𝑟 =
𝑛!
𝑛−𝑟 !
 With 𝑛! = 𝑛 ∗ 𝑛 − 1 ∗ 𝑛 − 2 ∗ ⋯ ∗ 3 ∗ 2 ∗ 1
 Example: 5! = 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 = 120
 Example: 7! = 7 ∗ 6 ∗ 5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 = 5,040
Permutations
 The number of ways to arrange 7 people or objects
taken 4 at a time is written 𝑃(7,4).
 𝑃 𝑛, 𝑟 =
 𝑃 7,4 =
𝑛!
𝑛−𝑟 !
Permutations
 The number of ways to arrange 7 people or objects
taken 4 at a time is written 𝑃(7,4).
 𝑃 𝑛, 𝑟 =
 𝑃 7,4 =
𝑛!
𝑛−𝑟 !
7!
7−4 !
Permutations
 The number of ways to arrange 7 people or objects
taken 4 at a time is written 𝑃(7,4).
 𝑃 𝑛, 𝑟 =
 𝑃 7,4 =
𝑛!
𝑛−𝑟 !
7!
7−4 !
=
7!
3!
Permutations
 The number of ways to arrange 7 people or objects
taken 4 at a time is written 𝑃(7,4).
 𝑃 𝑛, 𝑟 =
 𝑃 7,4 =
𝑛!
𝑛−𝑟 !
7!
7−4 !
=
7!
3!
=
7∗6∗5∗4∗3∗2∗1
3∗2∗1
Permutations
 The number of ways to arrange 7 people or objects
taken 4 at a time is written 𝑃(7,4).
 𝑃 𝑛, 𝑟 =
 𝑃 7,4 =
𝑛!
𝑛−𝑟 !
7!
7−4 !
=
7!
3!
=
7∗6∗5∗4∗3∗2∗1
3∗2∗1
=
Permutations
 The number of ways to arrange 7 people or objects
taken 4 at a time is written 𝑃(7,4).
 𝑃 𝑛, 𝑟 =
 𝑃 7,4 =
𝑛!
𝑛−𝑟 !
7!
7−4 !
=
7!
3!
=
7∗6∗5∗4∗3∗2∗1
3∗2∗1
=7∗6∗5∗4
Permutations
 The number of ways to arrange 7 people or objects
taken 4 at a time is written 𝑃(7,4).
 𝑃 𝑛, 𝑟 =
 𝑃 7,4 =
𝑛!
𝑛−𝑟 !
7!
7−4 !
=
7!
3!
=
7∗6∗5∗4∗3∗2∗1
3∗2∗1
 As was computed earlier.
= 7 ∗ 6 ∗ 5 ∗ 4 = 840
Example – Figure Skating
 There are 7 finalist for the world championship of
skating. How many ways can gold, silver and bronze
medals be awarded?
Example – Figure Skating
 There are 7 finalist for the world championship of
skating. How many ways can gold, silver and bronze
medals be awarded?
 Does order matter?
Example – Figure Skating
 There are 7 finalist for the world championship of
skating. How many ways can gold, silver and bronze
medals be awarded?
 Does order matter? YES!!! This is a permutation.
Example – Figure Skating
 There are 7 finalist for the world championship of
skating. How many ways can gold, silver and bronze
medals be awarded?
 Does order matter? YES!!! This is a permutation.
 𝑃 𝑛, 𝑟 =
Example – Figure Skating
 There are 7 finalist for the world championship of
skating. How many ways can gold, silver and bronze
medals be awarded?
 Does order matter? YES!!! This is a permutation.
 𝑃 𝑛, 𝑟 = 𝑃(7,3)
Example – Figure Skating
 There are 7 finalist for the world championship of
skating. How many ways can gold, silver and bronze
medals be awarded?
 Does order matter? YES!!! This is a permutation.
 𝑃 𝑛, 𝑟 = 𝑃(7,3)
 𝑃 7,3 =
7!
7−3 !
=
7!
4!
=
7∗6∗5∗4∗3∗2∗1
4∗3∗2∗1
= 7 ∗ 6 ∗ 5 = 210
Permutations with Repetition
 The number of permutations of 𝑛 objects of which 𝑝
are alike and 𝑞 are alike is
𝑛!
𝑝!𝑞!
 Example: How many permutations of the word POPPY
exist?
Permutations with Repetition
 The number of permutations of 𝑛 objects of which 𝑝
are alike and 𝑞 are alike is
𝑛!
𝑝!𝑞!
 Example: How many permutations of the word POPPY
exist?
 For this example 𝑛 = 5, 𝑝 = 3, 𝑞 = 1, 𝑟 = 1
# of P
# of O
# of Y
Permutations with Repetition
 The number of permutations of 𝑛 objects of which 𝑝
are alike and 𝑞 are alike is
𝑛!
𝑝!𝑞!
 Example: How many permutations of the word POPPY
exist?
# of P
# of O
# of Y
 For this example 𝑛 = 5, 𝑝 = 3, 𝑞 = 1, 𝑟 = 1
 𝑃𝑒𝑟𝑚𝑢𝑎𝑡𝑖𝑜𝑛𝑠 =
5!
3!∗1!∗1!
=
Permutations with Repetition
 The number of permutations of 𝑛 objects of which 𝑝
are alike and 𝑞 are alike is
𝑛!
𝑝!𝑞!
 Example: How many permutations of the word POPPY
exist?
# of P
# of O
# of Y
 For this example 𝑛 = 5, 𝑝 = 3, 𝑞 = 1, 𝑟 = 1
 𝑃𝑒𝑟𝑚𝑢𝑎𝑡𝑖𝑜𝑛𝑠 =
5!
3!∗1!∗1!
=
5∗4∗3∗2∗1
3∗2∗1∗1∗1
Permutations with Repetition
 The number of permutations of 𝑛 objects of which 𝑝
are alike and 𝑞 are alike is
𝑛!
𝑝!𝑞!
 Example: How many permutations of the word POPPY
exist?
# of P
# of O
# of Y
 For this example 𝑛 = 5, 𝑝 = 3, 𝑞 = 1, 𝑟 = 1
 𝑃𝑒𝑟𝑚𝑢𝑎𝑡𝑖𝑜𝑛𝑠 =
5!
3!∗1!∗1!
=
5∗4∗3∗2∗1
3∗2∗1∗1∗1
= 5 ∗ 4 = 20
Permutations with Repetition
 How many distinct permutations of the word MAMA
exist?
 Of the word MISSISSIPPI?
Permutations with Repetition
 How many distinct permutations of the word MAMA
exist?
 𝑃𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛𝑠 =
4!
2!∗2!
=
4∗3∗2∗1
2∗1∗2∗1
 Of the word MISSISSIPPI?
=
4∗3
2∗1
=
12
2
=6
Permutations with Repetition
 How many distinct permutations of the word MAMA
exist?
 𝑃𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛𝑠 =
4!
2!∗2!
=
4∗3∗2∗1
2∗1∗2∗1
=
4∗3
2∗1
=
12
2
=6
 Of the word MISSISSIPPI?
 𝑃𝑒𝑟𝑚𝑢𝑡𝑎𝑡𝑖𝑜𝑛𝑠 =
11!
4!4!2!1!
=
11∗10∗9∗8∗7∗6∗5∗4∗3∗2∗1
4∗3∗2∗1∗4∗3∗2∗1∗2∗1∗1
= 34,650
CW 12.2 – Permutations