Transcript Counting Principles Review

Lesson 7-1
Warm-up
 You 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 Principle
If 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 Principle
Example: 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 Principle
Example: 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 Principle
Example: 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 1
It'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 2
In 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 3
A 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
Counting
Example:
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?!?!?
FavorableOutcomes
Pr obability 
Total Outcomes
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 Probability
 You 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
French Onion
Vegetable
Sandwiches
Chicken
Club
Grilled Cheese
Finding the Probability
Solution:
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
orders that inclde vegetable soup 3 1
P(vegetable soup) 
 
total possible lunch orders
6 2
Solving a multi-step problem
 Playing 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 problem
Step 1: List the favorable outcomes. There are 6:
1-1-1-1-1
2-2-2-2-2
3-3-3-3-3
4-4-4-4-4
5-5-5-5-5
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.
Favorable Outcomes
6
1
P(all the same) 


Total Outcomes
7, 776 1296
Let’s Practice
 You 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 Practice
 The 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 Practice
 A 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 Practice
 Abby 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 Practice
 A 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 Practice
 The 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