a horse race has the following horses running. how many different first, second and third place...
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A horse race has the following horses running. How many different first, second and third place results are possible:
MushroomPepperSausage
TomatoOnionAnchovies
A horse race has the following horses running. How many different first, second and third place results are possible:
1st 2nd 3rd
M P S
M P T
M P O
M P A
M S P
M S T
M S O
M S A
M T P
M T O
M T A
M T S
M O P
M O S
M O T
M O A
M A P
M A S
M A T
M A O
P M S
P M T
P M O
P M A
P S M
P S T
P S O
P S A
P T M
P T S
P T O
P T A
P O M
P O S
P O T
P O A
P A M
P A S
P A T
P A O
S M P
S M T
S M O
S M A
S P M
S P T
S P O
S P A
S T M
S T P
S T O
S T A
S O M
S O P
S O T
S O A
S A M
S A P
S A T
S A O
T M P
T M S
T M O
T M A
T P M
T P S
T P O
T P A
T S M
T S P
T S O
T S A
T O M
T O P
T O S
T O A
T A M
T A P
T A S
T A O
O M P
O M S
O M T
O M A
O P M
O P S
O P T
O P A
O S M
O S P
O S T
O S A
O T M
O T P
O T S
O T A
O A M
O A P
O A S
O A T
A M P
A M S
A M T
A M O
A P M
A P S
A P T
A P O
A S M
A S P
A S T
A S O
A T M
A T P
A T S
A T O
A O M
A O P
A O S
A O T
A restaurant offers the following toppings for their pizzas. How many different 3 topping pizzas could be ordered?
MushroomPepperSausage
TomatoOnionAnchovies
A restaurant offers the following toppings for their pizzas. How many different 3 topping pizzas could be ordered?
Topping Topping Topping
Mushroom Pepper SausageMushroom Sausage PepperPepper Mushroom SausagePepper Sausage MushroomSausage Mushroom PepperSausage Pepper Mushroom
M P T
M P O
M P A
M S T
M S O
M S A
M T P
M T O
M T A
M T S
M O P
M O S
M O T
M O A
M A P
M A S
M
M
A
P S
M S P
T
M A O
P M T
P M O
P M A
P S T
P S O
P S A
P T M
P T S
P T O
P T A
P O M
P O S
P O T
P O A
P A M
P A S
P
P
A
M S
P S M
T
P A O
S M T
S M O
S M A
S P T
S P O
S P A
S T M
S T P
S T O
S T A
S O M
S O P
S O T
S O A
S A M
S A P
S
S
A
M P
S P M
T
S A O
T M P
T M S
T M O
T M A
T P M
T P S
T P O
T P A
T S M
T S P
T S O
T S A
T O M
T O P
T O S
T O A
T A M
T A P
T A S
T A O
O M P
O M S
O M T
O M A
O P M
O P S
O P T
O P A
O S M
O S P
O S T
O S A
O T M
O T P
O T S
O T A
O A M
O A P
O A S
O A T
A M P
A M S
A M T
A M O
A P M
A P S
A P T
A P O
A S M
A S P
A S T
A S O
A T M
A T P
A T S
A T O
A O M
A O P
A O S
A O T
M P O
M P A
M S T
M S O
M S A
M T O
M T A
M T S
M O P
M O S
M O T
M O A
M
M P T
A P
M
M
A
P S
S
M
M
A
M T P
S P
T
M A O
P M O
P M A
P S T
P S O
P S A
P T S
P T O
P T A
P O M
P O S
P O T
P O A
P
P M T
A M
P
P
A
M S
S
P
P
A
P T M
S M
T
P A O
S M T
S M O
S M A
S P T
S P O
S P A
S T M
S T P
S T O
S T A
S O M
S O P
S O T
S O A
S A M
S A P
S
S
A
M P
S P M
T
S A O
T M S
T M O
T M A
T P S
T P O
T P A
T S M
T S P
T S O
T S A
T O M
T O P
T O S
T O A
T A M
T A P
T
T
A
M P
T P M
S
T A O
O M P
O M S
O M T
O M A
O P M
O P S
O P T
O P A
O S M
O S P
O S T
O S A
O T M
O T P
O T S
O T A
O A M
O A P
O A S
O A T
A M P
A M S
A M T
A M O
A P M
A P S
A P T
A P O
A S M
A S P
A S T
A S O
A T M
A T P
A T S
A T O
A O M
A O P
A O S
A O T
There are 3! combinations of the same three toppings
A restaurant offers a three meal combination package for $10.00. How many different breakfast, lunch, dinner packages are possible?
Breakfast1. Mushroom2. Pepper3. Sausage
Lunch1. Ham2. Tomato
Dinner:1. Onion2. Olive3. Anchovies
Fundamental Counting Principle
A horse race has the following horses running. How many different first, second and third place results are possible:
MushroomPepperSausageHam
TomatoOnionOliveAnchoviesPermutation
A restaurant offers the following toppings for their pizzas. How many different 3 topping pizzas could be ordered?
MushroomPepperSausageHam
TomatoOnionOliveAnchoviesCombination
Permutations Combinations
The number of ways of arranging n items, taken r at a time is:
( , )n rP P n r
!
!
n
n r
The number of ways of combining n items, taken r at a time is:
( , )n rC C n r
!
! !
n
n r r
Evaluate Each Permutation
7 2
5 5
( )
( ) (8,5)
( )
( ) (7,1)
a P
b P
c P
d P
Evaluate Each Permutation
7 2
5 5
7! 7 6 5 4 3 2 17 6 42
(7 2)! 5 4 3 2 1
8! 8 7 6 5 4 3
(
2 18 7 6 5 4 6720
(8 5)! 3 2 1
5! 5 4 3 2 18 7 6 5 4 120
(5 5)! 1
7! 7 6 5 4 3 2 17
(7 1)! 6 5
)
( ) (8,5)
( )
4 3( ) 7,
1
2( 1)
a P
b P
c P
d P
Evaluate Each Combination
7 2
5 5
( )
( ) (8,5)
( )
( ) (7,1)
a C
b C
c C
d C
Evaluate Each Combination
7 2
5 5
7! 7 6 5 4 3 2 1 7 6 4221
(7 2)! 2! 5 4 3 2 1 2 1 2 2
8! 8 7 6 5 4 3 2 156
(8 5)! 5! 3 2 1 5 4 3 2 1
5! 5 4 3 2 11
(5 5)! 5! 1 5 4 3 2 1
7! 7 6 5 4 3 2 1
( )
( ) C(8,5)
( )
( ) C(7,1)(7 1)! 1! 6 5
a C
b
c C
d
7
4 3 2 1 1
How many ways can a 5 member club elect a President and Vice President?
Systematic Listing of Club Members: Aaron, Bob, Carly, Debbie, Eddie
Is this a permutation problem or a combination problem?
This is a PERMUTATION problem
How many ways can a 5 member club elect a President and Vice President?
Out of 5 members of a club, we are choosing 2 members, one to be President and the other to be VP
5 2 (5,2) 20 waysP P
How many ways can a 5 member club elect a two person committee?
Systematic Listing of Club Members: Aaron, Bob, Carly, Debbie, Eddie
Is this a permutation problem or a combination problem?
This is a COMBINATION problem
How many ways can a 5 member club elect a two person committee?
Out of a 5 member club, we are choosing 2 to be on a committee
5 2 (5,2) 10 waysC C
How do we determine if a problem is a permutation or a combination?
If the order in which items are chosen matters, that is the same items in a different order is a different outcome, then the problem is a PERMUTATION problem
How do we determine if a problem is a permutation or a combination?
If the order in which items are chosen DOES NOT matter, that is the same items in a different order is still the same outcome, then the problem is a COMBINATION problem
Is the combination AB the same as BA? Why or why not?
ABACADAE
BA
BCBDBE
CACB
CDCE
DADBDC
DE
EAEBECED
We could also use a product table to show the possible results.
ABACADAE
BA
BCBDBE
CACB
CDCE
DADBDC
DE
EAEBECED
Vic
e P
resi
dent
President
A B C D E
A B C D E
Another option would be to draw a tree diagram. Vice President
PresidentB C D EA
B
C
D
E