Counting PrinciplesLesson 7-1

Warm-upYou are at a restaurant with a special for

$10. You have the option to get: a) an appetizer and an entree or b) an entree and a dessert. There are 3 options for appetizers, 4 options for entrees, and 2 options for dessert. How many different meals can you make out of option a? Out of option b? How many total different meals can you make with the special?

Addition Counting PrincipleIf the possibilities being counted can be divided

into groups with no possibilities in common, then the total number of possibilities is the sum of the numbers of possibilities in each group.

If the possibilities being counted have common possibilities, P(A or B) = P(A) + P(B) – P(A and B)

Addition Counting PrincipleExample: For lunch in the cafeteria, you can

either have a milk or a juice with your meal. If there are 3 different types of milk and 4 types of juice, how many different choices do you have?

Addition Counting PrincipleExample: You are playing crazy eights and

could put down either a 5 or a heart. How many different cards will satisfy these conditions?

Multiplication Counting Principle

If one event can occur in m ways and another event can occur in n ways, then the number of ways that both events can occur together is m x n. This principle can be extended to 3 or more events.

Multiplication Counting PrincipleExample: The drama club is holding tryouts for

a play. With six men and eight women auditioning for the leading roles, how many

different couples could be made?

Question 1It's election time at school. 5 people are running for president, 3 people are running for vice-president, and 6 people are running for treasurer. How many different combinations of officers could you have in charge of your class?

Question 2In a town, each person gets a different telephone number. If no telephone number can start with a 0 or a 1, how many different 7-digit telephone numbers can be registered in that town?

Question 3A alpha-numeric code can either be 2 digits (numbers) followed by 1 letters or 2 letters followed by 1 digit. How many different codes can you make?

Use the Addition Principle of CountingExample:

Every purchase made on a company’s website is given a randomly generated confirmation code. The code consists of 3 symbols (letters and digits). How many codes can be generated if at least one letter is used in each?

Solution:To find the number of codes, find the sum of the numbers of possibilities for 1-letter codes, 2-letter codes, and 3-letter codes.

1-letter: There are 26 choices for each letter and 10 choices foe each digit. So there are 26x10x10 = 2,600 letter-digit-digit possibilities. The letter can be in any position so there are 3x2,600 = 7,800 possibilities.

2 letter: There are 26x26x10 = 6,760 letter-letter-digit possibilities. The digit can be in any of the three positions, so 3x6760 = 20,280 possibilities.

3 letter: There are 26x26x26 = 17, 576 letter-letter-letter possibilities.

So there are 7,800 + 20,280 + 17,576 = 45,656 possible codes

Let’s Recall…What is Probability?!?!?

PrFavorableOutcomes

obabilityTotalOutcomes

How does this apply to probability?Let’s take another look one of the last

problems:An alpha-numeric code can either be 2 digits (numbers) followed by 1 letter or 2 letters followed by 1 digit. What is the probability that if you picked a code at random, it would end with a “Z”?

Finding the ProbabilityYou are having lunch at a restaurant. You

order the special from the menu shown. If you randomly choose the soup and sandwich, what is the probability that your order includes vegetable soup.

Lunch Special $5.95

Choose 1 soup and 1 sandwich

Soups Sandwiches

French Onion Chicken

Vegetable Club

Grilled Cheese

Finding the ProbabilitySolution:

Because there are 2 soup choices and 3 sandwich choices, the total number of possible lunch orders is 2x3 = 6 . If you limit yourself to only 1 soup, vegetable soup, then the number of orders that include vegetable soup 1x3 = 3

3 1( )

6 2

orders that incldevegetable soupP vegetable soup

total possible lunchorders

Solving a multi-step problemPlaying a game, you and four friends each

roll a six-sided number cube. What is the probability that you each roll the same number?

Solving a Multi-step problemStep 1: List the favorable outcomes. There are 6:1-1-1-1-1 2-2-2-2-2 3-3-3-3-3 4-4-4-4-45-5-5-5-5 6-6-6-6-6

Step 2: Find the total number of outcomes using the multiplication principle. Total number of outcomes = 6x6x6x6x6 = 7,776

Step 3: Find the probability.

6 1( )

7,776 1296

FavorableOutcomesP all the same

TotalOutcomes

Let’s PracticeYou walk into an ice cream store…

There are 40 different types of ice cream and 20 different types of soft drinks. You are only allowed to buy one item from the store. How many different choices do you have for either ice cream or soft drinks?

40+20 = 60

Let’s PracticeThe same ice cream store has 40 different

types of ice cream…and they also have a choice of 3 different toppings!! How many different combinations of ice cream and toppings can you have from this ice cream store?

Using the multiplication principle of counting

we get 120 different choices.

Let’s PracticeA Chinese restaurant serves 25 rice dishes

and 10 noodle dishes. If Simon orders either a rice dish or a noodle dish, from how many dishes can he choose?

25 + 10 = 35

Let’s PracticeAbby has 3 hats, 4 scarves, and 3 pairs of

gloves. In how many different ways can she wear a hat, a scarf, and a pair of gloves?

3 * 4 * 3 = 36

Let’s PracticeA pirate walk into a bar…he has a choice of 3

different places to sit, 4 different items to drink, and 10 different items to eat. How many different combinations does the pirate have of sitting, drinking, and eating?

3*4*10 = 120

Let’s PracticeThe cafeteria serves 8 dishes that have meat

in them and 4 that do not. How many different choices of food would you have if you were to buy lunch from this cafeteria?

8+4 = 12