CombinationsCombinations

Finding All Possible OutcomesFinding All Possible Outcomes

What’s for Dessert?What’s for Dessert?

For dessert, Sam has a choice of two For dessert, Sam has a choice of two kinds of pie (apple or peach) and three kinds of pie (apple or peach) and three toppings (whipped cream, vanilla ice toppings (whipped cream, vanilla ice cream, and chocolate ice cream).cream, and chocolate ice cream).

How many different desserts are How many different desserts are possible?possible?

One way to find all the possible One way to find all the possible desserts is to make a desserts is to make a tree tree

diagramdiagram.. Sam has two choices of pies:Sam has two choices of pies:

apple peachapple peach

On each type of pie, he can put one On each type of pie, he can put one of three toppings:of three toppings:

Apple Peach

Whipped Whipped

cream cream

VanillaVanilla Vanilla Vanilla ice cream ice cream ice cream ice cream

On each type of pie, he can put one On each type of pie, he can put one of three toppings:of three toppings:

Apple Peach

Whipped Whipped

cream cream

VanillaVanilla Vanilla Vanilla ice cream ice cream ice cream ice cream

On each type of pie, he can put one On each type of pie, he can put one of three toppings:of three toppings:

Apple Peach

Whipped Whipped

cream cream

Chocolate Chocolate

ice cream ice cream

VanillaVanilla Vanilla Vanilla ice cream ice cream ice cream ice cream

How many different desserts are How many different desserts are possible?possible?

Apple Peach

Whipped Whipped

cream cream

Chocolate Chocolate

ice cream ice cream

What’s My Number?What’s My Number?

Sue has 4 cards on which one of the Sue has 4 cards on which one of the digits 2, 8, 5, or 4 is printed.digits 2, 8, 5, or 4 is printed.

How many different four-digit How many different four-digit numbers can she write using each of numbers can she write using each of the digits in each number?the digits in each number?

Another way to find all possible Another way to find all possible combinations is to combinations is to make an make an

organized listorganized list.. First list all the ways the numbers First list all the ways the numbers

could begin…could begin… 22 88 55 44

Then list each possible Then list each possible secondsecond digit….digit….

2288 2 255 2 244

8822 8 855 8 844

5522 5 588 5 54 4

4422 4 488 4 455

Next list each possible Next list each possible third third digit for each digit for each pair of numbers in the list….pair of numbers in the list….

228855 2 25588 2 24488 228844 2 25544 2 24455

882255 8 85522 8 84455 882244 8 85544 8 8442 2

552288 5 58822 5 54488 552244 5 58844 5 54422

442288 4 48822 4 4552 2 442255 4 48855 4 4558 8

Last, add the Last, add the fourthfourth digit to complete the digit to complete the four-digit numbers in the list….four-digit numbers in the list….

22885544 2 2558844 2 2448855 22884455 2 2554488 2 2445588

88225544 8 8552244 8 844552 2 88224455 8 855442 2 8 8442255

55228844 5 5882244 5 544882 2 55224488 5 588442 2 5 5442288

44228855 4 4882255 4 4552288 44225588 4 488552 2 4 455882 2

What patterns do you notice? What patterns do you notice? How many different combinations are there with 4 How many different combinations are there with 4

digits?digits? How many with 3 digits?How many with 3 digits?

22885544 2 2558844 22448855 22884455 2 2554488 2 2445588

88225544 8 8552244 8844552 2 88224455 8 855442 2 8 8442255

55228844 5 5882244 5544882 2 55224488 5 588442 2 55442288

44228855 4 4882255 4 4552288 44225588 4488552 2 4 455882 2

YOUR TURN!YOUR TURN!

Use a Use a tree diagramtree diagram or an or an organized organized listlist to show all possible three-letter to show all possible three-letter combinations using the letters combinations using the letters A, E, A, E, and Tand T..

How many three-letter combinations How many three-letter combinations are there?are there?

Which combinations form actual Which combinations form actual words?words?