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7-3 Multiplication Counting Principles
Holt Algebra 2
7-3 Multiplication Counting Principles
Example 1A: Using the Fundamental Counting
Principle
To make a yogurt parfait, you choose one
flavor of yogurt, one fruit topping, and one nut
topping. How many parfait choices are there?
Yogurt Parfait
(choose 1 of each)
Flavor
Plain
Vanilla
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Fruit
Peaches
Strawberries
Bananas
Raspberries
Blueberries
Nuts
Almonds
Peanuts
Walnuts
7-3 Multiplication Counting Principles
Example 1A Continued
number
number
number
number
equals
of
times of fruits times of nuts
of choices
flavors
2

5

There are 30 parfait choices.
Holt Algebra 2
3
=
30
7-3 Multiplication Counting Principles
Example 1B: Using the Fundamental Counting
Principle
A password for a site consists of 4 digits
followed by 2 letters. The letters A and Z are
not used, and each digit or letter many be used
more than once. How many unique passwords
are possible?
digit digit digit digit letter letter
10  10  10  10  24  24 = 5,760,000
There are 5,760,000 possible passwords.
Holt Algebra 2
7-3 Multiplication Counting Principles
Check It Out! Example 1a
A “make-your-own-adventure” story lets you
choose 6 starting points, gives 4 plot choices,
and then has 5 possible endings. How many
adventures are there?
number
of
starting
points
6


number
of plot
choices
4

number
of
possible
endings
=

5
=
There are 120 adventures.
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number
of
adventures
120
7-3 Multiplication Counting Principles
Check It Out! Example 1b
A password is 4 letters followed by 1 digit.
Uppercase letters (A) and lowercase letters (a)
may be used and are considered different. How
many passwords are possible?
Since both upper and lower case letters can be used,
there are 52 possible letter choices.
letter letter letter letter number
52
 52  52 
52 
10
= 73,116,160
There are 73,116,160 possible passwords.
Holt Algebra 2
7-3 Multiplication Counting Principles
Sample Space and Tree Diagrams
When attempting to determine a sample
space (the possible outcomes from an
experiment), it is often helpful to draw a
diagram which illustrates how to arrive at
the answer.
One such diagram is a tree diagram.
Holt Algebra 2
7-3 Multiplication Counting Principles
Sample Space and Tree Diagrams
• In addition to helping determine the number of
outcomes in a sample space, the tree diagram
can be used to determine the probability of
individual outcomes within the sample space.
• The probability of any outcome in the sample
space is the product (multiply) of all possibilities
along the path that represents that outcome on
the tree diagram.
Holt Algebra 2
7-3 Multiplication Counting Principles
Example 2
• Show the sample
space for tossing one
penny and rolling one
die.
(H = heads, T = tails)
Holt Algebra 2
7-3 Multiplication Counting Principles
Example 2 continued
• By following the different paths in the tree
diagram, we can arrive at the sample space.
• Sample space:
{ H1, H2, H3, H4, H5, H6,
T1, T2, T3, T4, T5, T6 }
• The probability of each of these outcomes is
1/2 • 1/6 = 1/12
• [The Counting Principle could also verify that this
answer yields the correct number of outcomes:
2 • 6 = 12 outcomes.]
Holt Algebra 2
7-3 Multiplication Counting Principles
Example 3
• A family has three
children. How many
outcomes are in the
sample space that
indicates the sex of
the children? Assume
that the probability of
male (M) and the
probability of female
(F) are each 1/2.
•
Holt Algebra 2
7-3 Multiplication Counting Principles
Example 3 continued
• Sample space:
•
{ MMM
MMF
MFM
MFF
FMM
FMF
FFM
FFF }
• There are 8 outcomes in the sample space.
• The probability of each outcome is
1/2 • 1/2 • 1/2 = 1/8.
Holt Algebra 2
7-3 Multiplication Counting Principles
Example 4
• A quiz has 10 “True or False” questions. If
you guess on each question, what is a
probability of getting each question right?
Holt Algebra 2
7-3 Multiplication Counting Principles
Selections with Replacement
• Let S be a set with n
elements. Then there
k
are n possible
arrangements of k
elements from S with
replacement.
Holt Algebra 2
7-3 Multiplication Counting Principles
Example 5
• Sarah decides to rank the five colleges
she plans on applying to. How many
rankings can she make?
Holt Algebra 2
7-3 Multiplication Counting Principles
Selections without Replacement
• Let S be a set with n elements. Then there
are n! possible arrangements of the n
elements without replacement.
Holt Algebra 2
7-3 Multiplication Counting Principles
Holt Algebra 2