Sec 4.4
Counting Rules
Bluman, Chapter 4
A Question to Ponder:
 A box contains 3 red chips, 2 blue chips and
5 green chips. A chip is selected, replaced
and a second chip is selected.
Display the sample space.
Do you think each event is
equally likely?
A Question to Ponder:
 A box contains 3 red chips, 2 blue chips and
5 green chips. A chip is selected, replaced
and a second chip is selected. Find the
following probabilities
a) Selecting 1 red chip and then 1 blue.
b) Selecting 1 blue chip and then 1 green chip.
c) Selecting 2 blue chips.
A Question to Ponder:
 A box contains 3 red chips, 2 blue chips and
5 green chips. A chip is selected, and not
replaced and a second chip is selected.
Display the sample space
Do you think each event is equally
likely?
A Question to Ponder:
 A box contains 3 red chips, 2 blue chips and
5 green chips. A chip is selected, and not
replaced and a second chip is selected. Find
the probability of each event
a) Selecting 1 red chip and then 1 blue chip.
b) Selecting 1 blue chip and then 1 green chip.
c) Matching chips.
Another Question to Ponder
 A game is played by drawing four cards from
an ordinary deck and replacing each card
after it is drawn. Find the probability of
winning if at least one ace is drawn.
4.4 Counting Rules
The
fundamental counting rule is also
called the multiplication of choices.
In
a sequence of n events in which the
first one has k1 possibilities and the second
event has k2 and the third has k3, and so
forth, the total number of possibilities of the
sequence will be
k1 · k2 · k3 · · · kn
Bluman, Chapter 4
Chapter 4
Probability and Counting Rules
Section 4-4
Example 4-39
Page #225
Bluman, Chapter 4
8
Example 4-39: Paint Colors
A paint manufacturer wishes to manufacture several
different paints. The categories include
Color: red, blue, white, black, green, brown, yellow
Type: latex, oil
Texture: flat, semigloss, high gloss
Use: outdoor, indoor
How many different kinds of paint can be made if you can
select one color, one type, one texture, and one use?

 
 
# of
# of
# of
# of
colors types textures uses
7

2

3

2
84 different kinds of paint
Bluman, Chapter 4
9
Example 4-40

There are four blood types, A, B, AB, and
O. Blood can also be Rh+ and Rh-.
Finally, a blood donor can be classified as
either male or female. How many different
ways can a donor have his or her blood
labeled?
Blood type ∙sign ∙gender
4∙2∙2=16
Example 4-41

The digits 1, 2, 3, 4, 5 and 6 are to be
used in a four-digit ID card. How many
different cards are possible if repetitions
are permitted?
First digit ∙ second digit ∙ third ∙ fourth
6∙
6 ∙
6 ∙
6= 1296
Counting Rules
 Factorial
is the product of all the positive
numbers from 1 to a number.
n !  n  n  1 n  2   3  2 1
0!  1
Bluman, Chapter 4
While you wait:
In how many ways can 3 people stand in
line at a store?
 In how many ways can 4 people be
chosen from a class of 12 to go on a FT?

Bluman, Chapter 4
Review topics
In how many ways can two items be chosen
from a selection of 4 . No replacement.
Bluman, Chapter 4
Consider the following cases
Ten runners compete in
an event.
The top 3 finishers
advance to the State
level.
Ten runners compete in
an event.
The first to cross the
finish line will receive a
gold medal, the second
place a silver and finally
the third finisher a
bronze medal.
Bluman, Chapter 4
Bluman, Chapter 4
There is 5-question
True or False test:
T=true; F=false
Are the following the
same result:
TFFTT
OR FFTTT

There are 5 DVDs.
Marge is sick and
staying home. In how
many ways can Marge
watch the DVDs.
Bluman, Chapter 4
Bluman, Chapter 4
Marge is shopping at
the local grocery store
and needs five items
from the produce
department. Jack is
also buying the same
five items.
Do you think Jack will
make his choices in the
same order as Marge?
Does it matter at the
end how they make
their choices?
There are 8 DVDs.
Marge is sick and
staying home. In how
many ways can Marge
watch the DVDs?
a) State when order
matters.
b) State when order
doesn’t matter.
Bluman, Chapter 4
Permutation Rule

The arrangement of n objects in a specific order
using r objects at a time is called a permutation
of n objects taking r objects at a time. It is
written as nPr and the formula is
n!
n pr 
(n  r )!
Counting Rules
Combination
is a grouping of objects.
Order does not matter.
n!
n Cr 
 n  r  !r !
Pr

r!
n
Bluman, Chapter 4
Permutation and combination
Order matters:
nPr
Order does NOT
matter
nCr
Bluman, Chapter 4
Chapter 4
Probability and Counting Rules
Section 4-4
Example 4-42/4-43
Page #228
Bluman, Chapter 4
23
Example 4-42: Business Locations
Suppose a business owner has a choice of 5 locations in
which to establish her business. She decides to rank
each location according to certain criteria, such as price
of the store and parking facilities. How many different
ways can she rank the 5 locations?





first second third fourth fifth
choice choice choice choice choice
5 
4  3  2  1

120 different ways to rank the locations
Using factorials, 5! = 120.
Using permutations, 5P5 = 120.
Bluman, Chapter 4
24
Example 4-43: Business Locations
Suppose the business owner in Example 4–42 wishes to
rank only the top 3 of the 5 locations. How many different
ways can she rank them?



first second third
choice choice choice
5  4  3

60 different ways to rank the locations
Using permutations, 5P3 = 60.
Bluman, Chapter 4
25
Chapter 4
Probability and Counting Rules
Section 4-4
Example 4-44
Page #229
Bluman, Chapter 4
26
Example 4-44: Television News Stories
A television news director wishes to use 3 news stories
on an evening show. One story will be the lead story, one
will be the second story, and the last will be a closing
story. If the director has a total of 8 stories to choose
from, how many possible ways can the program be set
up?
Since there is a lead, second, and closing story, we know
that order matters. We will use permutations.
8!
 336
8 P3 
5!
or
P  8  7  6  336
8 3
3
Bluman, Chapter 4
27
Chapter 4
Probability and Counting Rules
Section 4-4
Example 4-45
Page #229
Bluman, Chapter 4
28
Example 4-45: School Musical Plays
A school musical director can select 2 musical plays to
present next year. One will be presented in the fall, and
one will be presented in the spring. If she has 9 to pick
from, how many different possibilities are there?
Order matters, so we will use permutations.
9!
 72 or
9 P2 
7!
P  9  8  72
9 2
Bluman, Chapter 4
2
29
Chapter 4
Probability and Counting Rules
Section 4-4
Example 4-48
Page #231
Bluman, Chapter 4
30
Example 4-48: School Musicals
A newspaper editor has received 8 books to review. He
decides that he can use 3 reviews in his newspaper. How
many different ways can these 3 reviews be selected?
The placement in the newspaper is not mentioned, so
order does not matter. We will use combinations.
8!
 8!/  5!3!  56
8 C3 
5!3!
87 6
or 8C3 
 56 or
3 2
Bluman, Chapter 4
P3
 56
8C3 
3!
8
31
Chapter 4
Probability and Counting Rules
Section 4-4
Example 4-49
Page #231
Bluman, Chapter 4
32
Example 4-49: Committee Selection
In a club there are 7 women and 5 men. A committee of 3
women and 2 men is to be chosen. How many different
possibilities are there?
There are not separate roles listed for each committee
member, so order does not matter. We will use
combinations.
7!
5!
Women: 7C3 
 35, Men: 5C2 
 10
4!3!
3!2!
There are 35·10=350 different possibilities.
Bluman, Chapter 4
33
On your own:



Read:
TI 83 84 activities on
pages 208 & 235
Summary Table on
page 232



Answer:
Applying Concepts 44 on page 232
Exercises 4-4 starting
with # 1 every other
odd problem.
Bluman, Chapter 4