Counting PrinciplesCounting Principles Counting PrinciplesCounting Principles

Chapter 6.7Chapter 6.7

Buying a car• You can choose a Ford, Subaru, or

Porsche• You can choose a small or

medium car• You can choose red (R), purple

(P), white(W) or green (G)• How many choices do you have?

Objectives• Use the Fundamental Counting Principle

to find the number of ways 2 or more events can occur

• Find the number of ways a group of objects can be arranged in order

• Find the number of ways to choose several objects from a group without regard to order

• Use counting principles to find probabilities

The Fundamental Counting Principal

• If you have 2 events: 1 event can occur m ways and another event can occur n ways, then the number of ways that both can occur is m*n

• Event 1 = 4 types of meats• Event 2 = 3 types of bread• How many diff types of sandwiches can

you make?

• 4*3 = 12

3 or more events:• 3 events can occur m, n, & p ways, then

the number of ways all three can occur is m*n*p

• 4 meats• 3 cheeses• 3 breads• How many different sandwiches can you

make?• 4*3*3 = 36 sandwiches

The Fundamental Counting Principle

• If 1 event can occur in m ways and a second event can occur in n ways, the number of ways the 2 events can occur in sequence is m*n.

• This rule can be extended for any number of events occurring in sequence.

Buying a car• You can choose a Ford, Subaru, or

Porsche• You can choose a small or medium

car• You can choose red (R), purple (P),

white(W) or green (G)• How many choices do you have?• 3*2*3 = 18 choices

You can draw a tree diagram to show this

Tree Diagram• A tool for helping you organize the

number of possible outcomes

What is the probability of drawing a red ball and a blue ball from a sack

with a red, blue, and white ball.

Security Systems• The access code for a car’s security

system consists of 2 digits. Each digit can be 0 through 9.

• How many access codes are possible if each digit can be used only once and not repeated?

• 10*10• = 100

Security Systems• What is the probability that you

guess it on the first try?

Security Systems• The access code for a car’s security

system consists of 4 digits. Each digit can be 0 through 9.

• How many access codes are possible if each digit can be repeated?

• 10*10*10*10• = 10,000

License Plates• How many license plates can you

make if a license plate consists of• Six (out of 26) letters each of

which can be repeated?• Six (out of 26) letters each of

which can not be repeated?