# The Counting Principle (Multiplication Principle) Multiplication principle: the total number of outcomes for an event is found by multiplying the number.

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The Counting Principle (Multiplication Principle) Multiplication principle: the total number of outcomes for an event is found by multiplying the number of choices for each stage of the event The multiplication principle gives you the number of outcomes NOT the probability.

Examples: 1. At an ice cream shop there are 31 flavors and 25 toppings. How many different ways are there to make a one-scoop ice cream sundae with one topping? 31 ● 25 = 775 2. There are 4 quarterbacks and 6 centers on a football team that has 60 players. How many quarterback-center pairings are possible? 3. You roll three dice, how many outcomes are there? 6 ● 6 ● 6 = 216 4 ● 6 = 24

4. Three coins are tossed, how many outcomes are there? 2 ● 2 ● 2=8 5. Canned bean are packed in three sizes: small, medium and large; and are red, black-eyed, green, yellow, or baked. How many size-type outcomes are there? 3 ● 5=15 6. A confectioner offers milk, dark, or white chocolates with solid, cream, jelly, nut, fruit, or caramel centers. How many flavor choices are there? 3 ● 6=18

Tree Diagrams and Counting Jennifer and her family went to Dilly’s Deli for lunch. Jennifer wanted a sandwich. She had three choices of bread: white, wheat or rye. She had three choices of meat: turkey, ham or roast beef. She can choose one type of meat and one type of bread for her sandwich. Complete the tree diagram and find the number of choices that Jennifer has for her sandwich. Total choices = 3 breads ● 3 meats = 9 sandwiches

A Tree Diagram can be used to show you the number of outcomes in an event. 9 choices

Examples: Cheryl has a choice of a pink, red or yellow blouse with white or black slacks for an outfit. How many possible outfits are there? 6 outfits

A coin is tossed and a spinner is spun. How many outcomes are there? 1 2 3 6 outcomes

Permutations With your group find as many arrangements of the letters A, H, M, T as you can. How many 2 letter arrangements are there? Could you do this an easier way? Use a tree diagram, or…

Examples: How many different ways can the letters of each word be arranged? 1. SAND 2. GREEN 3. CAT 4! = 4 ● 3 ● 2 ● 1 = 24 5! = 5 ● 4 ● 3 ● 2 ● 1 = 120 3! = 3 ● 2 ● 1 = 6

Examples: Find the value. 4. 7! 5. P(8,2) 6. P(9, 3) 7 ● 6 ● 5 ● 4 ● 3 ● 2 ● 1 = 5040 8 ● 7 = 56 9 ● 8 ● 7 = 504

Examples: 7. In how many ways can six people line up for a photograph? 8. A building inspector is supposed to inspect 10 building for safety code violations. In how many different orders can the inspector visit the buildings? 6! =6●5●4●3●2●16●5●4●3●2●1 720 ways 10 ! =10 ● 9 ● 8 ● 7 ● 6 ● 5 ● 4 ● 3 ● 2 ● 1 = 3,628,800 ways

Examples: 9. How many 3 letter words can you make from 5 letters? 10. How many 4-letter, two digit license plate numbers can you make? P(5, 3) 5●4●35●4●3 60 words 26 ● 26 ● 26 ● 26 ● 10 ● 10 a. If repeat letters and numbers allowed b. If repeat letters and numbers not allowed 26 ● 25 ● 24 ● 23 ● 10 ● 9 45,697,600 plates 32,292,000 plates

Probability of AND events: Notation: P(A and B) = P(A)P(B) Probability of AND events, you MULTIPLY! If you draw a card from a deck numbers 1 through 10 and toss a die, find the probability of each outcome. Outcomes when you roll a die: Outcomes when you pick a card: Event A: pick a card Event B: roll a die 1, 2, 3, 4, 5, 6, 7, 8, 9, 101, 2, 3, 4, 5, 6

1. P(a 10 and a 3) 2. two even numbers If you draw a card from a deck numbers 1 through 10 and toss a die, find the probability of each outcome. Outcomes: 6Outcomes: 10 Event A: pick a cardEvent B: roll a die P(10) = _1_ 10 P(3) = _1_ 6 P(10 and 3)=P(2even)= _1_ _1_ = 10 6 _1_ 60 P(even) = _5_ = _1_ 10 2 P(even) = _1_ 2 _1_ _1_ = 2 _1_ 4

3. P(2primenumbers) 4. two odd numbers If you draw a card from a deck numbers 1 through 10 and toss a die, find the probability of each outcome. Outcomes: 6Outcomes: 10 Event A: pick a cardEvent B: roll a die P(prime) = _2_ 5 P(prime) = _3_ = _1_ 6 2 P(prime and prime)= P(2odd)= _2_ _1_ = 5 2 _1_ 5 P(odd) = _5_ 10 P(odd) = _1_ 2 _1_ _1_ = 2 _1_ 4

5. P(even and prime) If you draw a card from a deck numbers 1 through 10 and toss a die, find the probability of each outcome. Outcomes: 6Outcomes: 10 Event A: pick a cardEvent B: roll a die P(even) = _1_ 2 P(prime) = _3_ = _1_ 6 2 P(prime and prime)= _1_ _1_ = 2 _1_ 4

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