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THE
ICE
CREAM
SOCIAL
Caught Doing Something Good?JOIN US FOR AN ICE CREAM SOCIAL
Choose one flavor of ice cream Chocolate Vanilla Strawberry
C H O O S E T W 0 T O P P I N G S :
C H O C O L AT E S Y R U P
W H I P P E D C R E A M
N U T S
C H E R R I E S
P I N E A P P L E
THE ICE CREAM SOCIAL
Rewarded with an ice cream social, students may select one flavor of ice cream and two toppings.
There are three flavors of ice cream and five toppings to choose from. How many combinations of ice cream and toppings are possible?
What is the probability that you will correctly guess the combination a student will order?
TO VISUALIZE THE SAMPLE SPACE YOU CAN CREATE A TREE DIAGRAM
Chocolate
Vanilla
Strawberry
Choose a flavor of ice cream
TO VISUALIZE THE SAMPLE SPACE YOU CAN CREATE A TREE DIAGRAM
Chocolate
• Chocolate Syrup• Whipped Cream• Nuts• Cherries• Pineapple
Vanilla
• Chocolate Syrup• Whipped Cream• Nuts• Cherries• Pineapple
Strawberry
• Chocolate Syrup• Whipped Cream• Nuts• Cherries• Pineapple
Choose the first topping
TO VISUALIZE THE SAMPLE SPACE YOU CAN CREATE A TREE DIAGRAM
Chocolate Syrup
• Whipped Cream
• Nuts• Cherries• Pineapple
Whipped Cream
• Nuts• Cherries• Pineapple
Nuts• Cherries• Pineapple
Cherries • Pineapples
Pineapple
• No more unique combinations
Choose the second topping
Chocolate
4 choices
3 choices
2 choices
1 choices
0 choices
10 Total choices
TO VISUALIZE THE SAMPLE SPACE YOU CAN CREATE A TREE DIAGRAM
Chocolate Syrup
•Whipped Cream
•Nuts•Cherries•PineappleWhipp
ed Cream
•Nuts•Cherries•Pineapple
Nuts
•Cherries•Pineapple
Cherries
•Pineapples
Pineapple
•No more unique combinations
The same choices can be made for vanilla
Vanilla
TO VISUALIZE THE SAMPLE SPACE YOU CAN CREATE A TREE DIAGRAM
Chocolate Syrup
•Whipped Cream
•Nuts•Cherries•PineappleWhipp
ed Cream
•Nuts•Cherries•Pineapple
Nuts
•Cherries•Pineapple
Cherries
•Pineapples
Pineapple
•No more unique combinations.
and for strawberry, for a total of 3 x 10 or 30 choices.
Strawberry
DETERMINE THE SAMPLE SPACE
If we could use the multiplication rule for this problem, we would have.
3 Ice cream flavor s
x 5 choices for the 1st topping
x 4 choices for the 2nd topping
Total possible combinations is
3 x 5 x 4 = 60
However, the tree diagram reveals there are only 30 unique choices.
The probability of guessing the order correctly is 1/30.This problem involves the MULTIPLICATION RULE and a COMBINATION.
The number of combinations of 3 ice cream flavors and 5 toppings , taken 2 at a time, is3 x 5C2 = 3 x 5! = 3 x 5!___ = 3 x 5 x 4 x 3 x 2 x 1 = 30
2!(5-2)! 2!(3)! 2 x 1 x 3 x 2 x 1
MULTIPLICATION RULE AND COMBINATIONSSuppose students may choose a third topping . What is the
size of the new sample space.?
Rather than completing the tree, you can use the combination formula to calculate the size of the sample space.
3 Ice cream flavor s
x 5 toppings taken 3 at a time.
Total possible combinations is 30
The number of combinations of 3 ice cream flavors and 5 toppings , taken 3 at a time, is3 x 5C3 = 3 x 5! = 3 x 5!___ = 3 x 5 x 4 x 3 x 2 x 1 = 30
3!(5-3)! 3!(2)! 3 x 2 x 1 x 2 x 1
EXTENSION PROBLEM
Marble Slab is famous for its flavors and toppings. How many combinations are possible if the High School moves its ice cream social to Marble Slab.
To calculate the number of samples, you need to know the number of flavors and the number of toppings. Visit the Marble Slab site to find out. Assume all the flavors and toppings are available.
MARBLE SLAB
There are 67 ice cream flavors and 68 toppings.
If all the flavors and toppings are available,
The number of combinations of 67 ice cream flavors and 68 toppings , taken 2 at a time, is
67 x 68C2 = ?
A problem of this magnitudes calls for a little help from technology.
