Counting Principles
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Tree Diagrams
A tree diagram can help you determine how many ways
a task can be completed.
Example 1:
You go on vacation, you pack three pairs of shorts and four
tops. How many different outfits can you wear? (assume
all shorts are neutral).
There are 12 outfit possibilities
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Tree Diagram
Can you easily make a tree diagram to help you answer
the following questions?
Example 2:
A meal consists of a main dish, a side dish, and a dessert.
How many different meals can be selected if there are 4
main dishes, 2 side dishes and 5 desserts available?
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Fundamental Counting Principle
If one event can occur in m ways and a second event can
occur in n ways, the number of ways the two events can
occur in sequence is m· n.
Example 1:
You go on vacation, you pack three pairs of shorts and four
tops. How many different outfits can you wear? (assume
all shorts are neutral).
# of shorts
3
# of
outfits
# of tops

4
=
12
There are 12 outfit possibilities
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Fundamental Counting Principle
If one event can occur in m ways and a second event can
occur in n ways, the number of ways the two events can
occur in sequence is m· n. This rule can be extended
for any number of events occurring in a sequence.
Example 2:
A meal consists of a main dish, a side dish, and a dessert.
How many different meals can be selected if there are 4
main dishes, 2 side dishes and 5 desserts available?
# of main
# of side
dishes
dishes
4
2

There are 40 meals available.

# of
desserts
5
=
40
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Fundamental Counting Principle
Example 3:
Two coins are flipped. How many different outcomes are
there?
Assume the two coins, one is a dime the other a quarter
Start
1st Coin
Tossed
Heads
2 ways to flip the coin
Tails
2nd Coin
Tossed
Heads
Tails
Heads
Tails
2 ways to flip the coin
There are 2  2 = 4 different outcomes: {HH, HT, TH, TT}.
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Fundamental Counting Principle
Example 4:
An iPhone allows you to put a security code to limit others’
ability to access your data. The security code consists of 4
digits. Each digit can be 0 through 9. How many different
codes can be made if
a.) each digit can be repeated
b.) each digit can only be used once and not repeated
a.) Because each digit can be repeated, there are 10
choices for each of the 4 digits.
10 · 10 · 10 · 10 = 10,000 codes
b.) Because each digit cannot be repeated, there are 10
choices for the first digit, 9 choices left for the second
digit, 8 for the third, 7 for the fourth
10 · 9 · 8 · 7 = 5040 codes
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Fundamental Counting Principle
Example 5:
You are about to take a 10 question multiple choice test.
Each question has four possible answers. How many
different ways could you fill in your answer sheet
(assuming you only fill in exactly one answer choice per
question)?
4 · 4 · 4 · 4 · 4 · 4 · 4 · 4 · 4 · 4= 410 =1,048,576
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Fundamental Counting Principle
Example 6:
In designing a computer, if a byte is defined to be a
sequence of 8 bits and each bit must be a 0 or 1, how many
different bytes are possible?
2· 2 · 2 · 2 · 2 · 2 · 2 · 2 = 256
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Fundamental Counting Principle
Example 7:
You have 2 six-sided dice. One die is red, the other die is
blue. You roll the two dice. How many different outcomes
are possible?
6 · 6 = 36
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Fundamental Counting Principle
Example 8:
Seven students go to the movies, 2 boys and five girls. They
are able to find 7 seats together.
a.) How many different seating arrangements are there?
b.) How many different seating arrangements are there if
both of the boys must sit together and all of the girls must sit
together.
c.) How many different seating arrangements are there if
both genders are not permitted to sit together.
a.) 7 · 6 · 5 · 4 · 3 · 2 · 1 = 5,040
b.) 5 · 4 · 3 · 2 · 1 = 120 (ways girls can sit together)
2 · 1 = 2 (ways boys can sit together)
120 · 2 + 2 · 120 = 480
c.) 5,040-480=4,560
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Fundamental Counting Principle
Example 9:
When designing surveys, pollsters sometimes try to
minimize a lead-in effect by rearranging the order in which
the questions are presented. (A lead-in effect occurs when
some questions influence the responses to questions that
follow.) If Gallup plans to conduct a consumer survey by
asking subjects 5 questions, how many different versions of
the survey are required if all possible arrangements are
included?
5 · 4· 3 · 2 · 1=120
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Fundamental Counting Principle
Example 10:
There are five people competing in a race. How many
different possibilities are there for first second and third
place?
5•4•3=60
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Fundamental Counting Principle
Example 11:
In the 1940s, AT&T and Bell Laboratories developed the area
code system for North America. At the time they were
developed, area codes were designed to be three digits. The
first digit could not be a zero or a one and the second digit
had to be a zero or a one. Given those constraints, how many
different area codes were available?
8· 2· 10 = 160 codes
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Fundamental Counting Principle
Example 12:
Phone numbers in the United States are 7 digits (following
the area code). The first digit cannot be a zero or one. The
next two digits can be any number; however, they cannot
both be 1. The final four digits can be any number. How
many phone numbers per area code are possible?
HINT: Think of two possibilities, either the second digits is
1, OR the second digit is not 1.
8•9•10•10•10•10•10 + 8•1•9•10•10•10•10=7,920,000
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Fundamental Counting Principle
Example 13:
You have to create a four letter password for an online
account you are creating. Determine how many passwords
are possible given the following criteria.
a.) Each character has to be a letter.
b.) Each character has to be a letter and no letter can be
repeated.
c.) Each character has to be a letter and the password is case
sensitive.
a.) 26 · 26 · 26 · 26 = 456,976
b.) 26 · 25 · 24 · 23 = 358,800
c.) 52 · 52 · 52 · 52 = 7,311,616
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Fundamental Counting Principle
Example 14:
A paint manufacturer wishes to make several differetn
paints. The categories include
 COLOR: Red, blue, white, black, green, brown, yellow
 TYPE: Latex, oil
 TEXTURE: Flat, semigloss, high gloss
 USE: Outdoor, indoor
7•2•3•2=84
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Fundamental Counting Principle
Example 15:
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?
4•2•2=16
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Fundamental Counting Principle
Example 16:
Students are classified according to eye color (blue, brown,
green), gender (male, female), and major (chemistry,
mathematics, physics, business). How many possible
different classifications are there?
3•2•4=24
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Fundamental Counting Principle
Example 17:
According to the most recent school profile data, GCHS has
1,410 students. The student body includes 346 seniors, 334
juniors, 368 sophomores and 362 freshman. A committee is
to be formed that consists of one seniors and one junior.
How many different committees are possible?
346 · 334 = 115,564
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Fundamental Counting Principle
Example 18:
The senior class consists of approximately 346 individuals.
Four class officers make up the executive board, president,
vice president, treasurer and secretary. How many possible
executive boards can be made?
346 · 345 · 344 · 343 ≈ 14 billion
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Counting Methods
Example 19:
In each of the following words, how many unique ways can
the letters be arranged
a.) LAD
b.) DAD
a.) 3 · 2 · 1 = 6 different “words”
b.) (3 · 2 · 1)/(2· 1) = 3 “different words
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Permutations
The number of possible arrangements (order matters) of a
specific size from a group of objects is referred to as a
permutation.
The number of permutations of n elements taken r at
a time is
n Pr 
# in the
group
n! .
(n  r)!
# taken from
the group
Example:
You are required to read 5 books from a list of 8. In how
many different orders can you do so?
n Pr  8 P5 
8!  8! = 8  7  6  5  4  3  2  1  6720 ways
(8  5)! 3!
3  2 1
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Distinguishable Permutations
The number of distinguishable permutations of n objects,
where n1 are one type, n2 are another type, and so on is
n!
, where n1  n2  n3   nk  n.
n1 !  n2 !  n3 ! nk !
Example:
Jessie wants to plant 10 plants along the sidewalk in her
front yard. She has 3 rose bushes, 4 daffodils, and 3 lilies.
In how many distinguishable ways can the plants be
arranged?
10  9  8  7  6  5  4!
10!

3!4!3!
3!4!3!
 4,200 different ways to arrange the plants
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Counting Methods
Example 20:
In each of the following words, how many unique ways can
the letters be arranged
a.) CARD
b.) CALC
c.) SASSY
d.) REGRET
a.) 4 · 3 · 2 · 1 = 24 different “words”
b). (4 · 3 · 2 · 1 )/(2 · 1)=12 different “words”
c). (5· 4 · 3 · 2 · 1 )/(3· 2· 1)=20 different “words”
d). (6· 5· 4 · 3 · 2 · 1 )/[(2· 1)· (2 · 1)]=180 different “words”
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Permutations
Some of the examples we completed earlier can be
described as permutations or distinguishable
permutations.
HOMEWORK: finish worksheet and identify any
examples that could be described as permutations or
distinguishable permutations.
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