Creating Combinations 6.4.1.1 Determine the sample space (set of possible outcomes) for a given experiment and determine which members of the sample space are related to certain events. Sample space may be determined by the use of tree diagrams, tables, or pictorial representations.
Finding Outcomes 13.2 Notes Methods Tree Diagram Table Organized List Combinations – Order is NOT important Permutations– Order is important Methods Tree Diagram Table Organized List
I can… Self Assessment Use tree diagrams to find outcomes Use tables or lists to find outcomes Self Assessment 5- I can do it without help & teach others. 4- I can do this with no help, but I don’t know if I can explain it. 3- I can do this with a little help. 2- I can do this with a lot of help! 1- I don’t have a clue.
What are creating combination problems? Creating combination problems are stories that ask you to find all the possible matches, combinations, pairs, or outecomes that can be made out of a group of objects.
For example, we have all seen problems like this: Suzanne has a red shirt, a blue shirt, and a pink shirt. She also has a pair of black pants and a pair of gray pants. How many different ways can Suzanne wear one shirt and one pair of pants?
x Wrong Answer! There’s the outfit I like. I’m done. Some students It is important to remember that you are not just making one outfit for Suzanne, but you are finding all the different outfits she could wear. You are determining all the different ways she could match up the shirts she has with the pants she has. There’s the outfit I like. I’m done. Some students make the mistake of putting just the answer they like best and not finishing the problem. Wrong Answer!
Don’t be tricked by the word “one” in the problem. That doesn’t mean make only one outfit. That just means Suzanne can only wear one shirt with one pair of pants at a time. Don’t try to dress her in three shirt and two pants all at once. That would look pretty silly!!!
red shirt with black pants blue shirt with black pants So, how do we find out how many outfits Suzanne has to choose from? There is an easy way and a hard way. The hard way is to just randomly match up shirts and pants, write them down, and hope we find them all without accidentally repeating any. red shirt with black pants blue shirt with black pants pink shirt with gray pants
? ? ? ? ? ? ? ? ? But how do we know for certain that we have found all the possible combinations or matches? ? ? ? ? ?
The Easy Way!! The hard way can get pretty confusing and it can take a long time to figure out. But…. Smart mathematicians know The Easy Way!!
red shirt black pants blue shirt gray pants pink shirt The easy method is called a Tree Diagram. Step 1: Make simple lists of the items you are matching. red shirt blue shirt pink shirt black pants gray pants
black pants red shirt gray pants black pants blue shirt gray pants Step 2: Build your trees. The first list will be the trunks and the second list is the branches. Suzanne can wear her red shirt with the black pants or she can wear the red shirt with her gray pants. So far that’s 2 different combinations. black pants red shirt gray pants black pants blue shirt 2 more gray pants black pants And 2 more. pink shirt gray pants 6 outfits in all!
3 shirts that can be worn with 2 pants each = 6 outfits Suzanne can use her three shirts and two pairs of pants to make six different outfits. 3 shirts that can be worn with 2 pants each = 6 outfits 3 x 2 = 6
Example 2 – a fair coin is flipped twice, what are all the possible outcomes? 1st 2nd H HH Possible Outcomes H T HT H TH T T TT
Example 3 – Your art class is painting pottery for an art fair Example 3 – Your art class is painting pottery for an art fair. You can choose a small or a large size and you can paint a vase, a jar, or a plate, What are the different kids of pottery you can paint? Step 1: List the sizes S L
Step 2: List the items for each size Example 3 – Your art class is painting pottery for an art fair. You can choose a small or a large size, and you can paint a vase, a jar, or a plate, What are the different kids of pottery you can paint? Step 2: List the items for each size V J S P V L J P
Step 2: Find the Outcomes Example 3 – Your art class is painting pottery for an art fair. You can choose a small or a large size, and you can paint a vase, a jar, or a plate, What are the different kids of pottery you can paint? Step 2: Find the Outcomes V SV Possible Outcomes J SJ S P SP V LV L J LJ P PT
Your Turn– A roller hockey team is choosing jerseys Your Turn– A roller hockey team is choosing jerseys. The body can be red, white, purple, or green. The sleeves can be black, red, or blue. List all the possible jerseys that the team can choose. Red Black Black R Red Red Red Blue Red Blue Black White Black W White Red Red Blue White Blue Possible Outcomes Black Purple Black P Red Purple Red Purple Blue Blue Green Black Black Green Red G Red Green Blue Blue
Combinations vs. Permutations Combination group of objects in which order is NOT important Example: Choosing 2 pizza toppings from a list of 5 or flipping a coin Permutation group of objects in which order is important Example: Order of doing chores or finding all of the 2 digit numbers from a list of 5 digits
Are you finding an combination or a permutation? Using a table You can choose 2 toppings for a sundae from nuts, sprinkles, caramel, and marshmallows. Find all the possible pairs of toppings. Are you finding an combination or a permutation? It doesn’t matter which topping is selected first.
Using a table You can choose 2 toppings for a sundae from nuts, sprinkles, caramel, and marshmallows. Find all the possible pairs of toppings. Nuts Sprinkles Caramel Marshmallows Outcomes
Using a table You can choose 2 toppings for a sundae from nuts, sprinkles, caramel, and marshmallows. Find all the possible pairs of toppings. Nuts Sprinkles Caramel Marshmallows Outcomes X NUTS, SPRINKLES
Using a table You can choose 2 toppings for a sundae from nuts, sprinkles, caramel, and marshmallows. Find all the possible pairs of toppings. Nuts Sprinkles Caramel Marshmallows Outcomes X NUTS, SPRINKLES NUTS, CARAMEL
Using a table You can choose 2 toppings for a sundae from nuts, sprinkles, caramel, and marshmallows. Find all the possible pairs of toppings. Nuts Sprinkles Caramel Marshmallows Outcomes X NUTS, SPRINKLES NUTS, CARAMEL NUTS, MARSHMALLOW
Using a table You can choose 2 toppings for a sundae from nuts, sprinkles, caramel, and marshmallows. Find all the possible pairs of toppings. Nuts Sprinkles Caramel Marshmallows Outcomes X NUTS, SPRINKLES NUTS, CARAMEL NUTS, MARSHMALLOW SPRINKLES, CARAMEL
SPRINKLES, MARSHMALLOW Using a table You can choose 2 toppings for a sundae from nuts, sprinkles, caramel, and marshmallows. Find all the possible pairs of toppings. Nuts Sprinkles Caramel Marshmallows Outcomes X NUTS, SPRINKLES NUTS, CARAMEL NUTS, MARSHMALLOW SPRINKLES, CARAMEL SPRINKLES, MARSHMALLOW
SPRINKLES, MARSHMALLOW Using a table You can choose 2 toppings for a sundae from nuts, sprinkles, caramel, and marshmallows. Find all the possible pairs of toppings. Nuts Sprinkles Caramel Marshmallows Outcomes X NUTS, SPRINKLES NUTS, CARAMEL NUTS, MARSHMALLOW SPRINKLES, CARAMEL SPRINKLES, MARSHMALLOW CARAMEL, MARSHMALLOW
Your Turn You can choose 2 vegetables for a stir-fry from green peppers, pea pods, onions, and broccoli. List all the possible pairs of vegetables. Green peppers Pea pods Onions Broccoli Outcomes
Your Turn You can choose 2 vegetables for a stir-fry from green peppers, pea pods, onions, and broccoli. List all the possible pairs of vegetables. Green peppers Pea pods Onions Broccoli Outcomes X Green pepper, pea pod
Your Turn You can choose 2 vegetables for a stir-fry from green peppers, pea pods, onions, and broccoli. List all the possible pairs of vegetables. Green peppers Pea pods Onions Broccoli Outcomes X Green pepper, pea pod Green pepper, onions
Your Turn You can choose 2 vegetables for a stir-fry from green peppers, pea pods, onions, and broccoli. List all the possible pairs of vegetables. Green peppers Pea pods Onions Broccoli Outcomes X Green pepper, pea pod Green pepper, onions Green pepper, broccoli
Your Turn You can choose 2 vegetables for a stir-fry from green peppers, pea pods, onions, and broccoli. List all the possible pairs of vegetables. Green peppers Pea pods Onions Broccoli Outcomes X Green pepper, pea pod Green pepper, onions Green pepper, broccoli Pea pods, onion
Your Turn You can choose 2 vegetables for a stir-fry from green peppers, pea pods, onions, and broccoli. List all the possible pairs of vegetables. Green peppers Pea pods Onions Broccoli Outcomes X Green pepper, pea pod Green pepper, onions Green pepper, broccoli Pea pods, onion Pea pods, broccoli
Your Turn You can choose 2 vegetables for a stir-fry from green peppers, pea pods, onions, and broccoli. List all the possible pairs of vegetables. Green peppers Pea pods Onions Broccoli Outcomes X Green pepper, pea pod Green pepper, onions Green pepper, broccoli Pea pods, onion Pea pods, broccoli Onions, broccoli
Making an Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Are you finding an combination or a permutation? Each outcome is a permutation because the order of the digits matters.
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17 19
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17 19 Start with 4
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17 19 Start with 4 41
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17 19 Start with 4 41 47
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17 19 Start with 4 41 47 49
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17 19 Start with 4 41 47 49 Start with 7
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17 19 Start with 4 41 47 49 Start with 7 71
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17 19 Start with 4 41 47 49 Start with 7 71 74
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17 19 Start with 4 41 47 49 Start with 7 71 74 79
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17 19 Start with 4 41 47 49 Start with 7 71 74 79 Start with 9
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17 19 Start with 4 41 47 49 Start with 7 71 74 79 Start with 9 91
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17 19 Start with 4 41 47 49 Start with 7 71 74 79 Start with 9 91 94
Organized List List all two digit numbers that can be formed using two different digits from 1, 4, 7, and 9. Start with 1 14 17 19 Start with 4 41 47 49 Start with 7 71 74 79 Start with 9 91 94 97
Your Turn You are placing a math book, a novel, and a dictionary on a shelf. List all possible ways you can order the books on the shelf. Math book, novel, dictionary Math book, dictionary, novel Novel, math book, dictionary Novel, dictionary, math book Dictionary, math book, novel Dictionary, novel, math book
I can… Self Assessment Use tree diagrams to find outcomes Use tables or lists to find outcomes Self Assessment 5- I can do it without help & teach others. 4- I can do this with no help, but I don’t know if I can explain it. 3- I can do this with a little help. 2- I can do this with a lot of help! 1- I don’t have a clue.
Finding Outcomes 13.2 Notes Methods Tree Diagram Table Organized List Combinations – Order is NOT important Permutations– Order is important Methods Tree Diagram Table Organized List