The Fundamental Counting Principle
and Permutations
What are all of the different scores you can get from a die roll
(one die) and a spin of the following spinner?
The Fundamental Counting Principle
and Permutations
As an area model, your solution might
look like this:
There are 48
possible
outcomes
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
7
8
9
10
11
12
13
8
9
10
11
12
13
14
The Fundamental Counting Principle
and Permutations
What if you were to add in a coin toss?
How many different outcomes could there be?
The Fundamental Counting Principle
If one event can occur in m ways and the
other event can occur in n ways, then the
number of ways that both can occur is m n
In our example with the die and spinner this
would be 6 8 or 48 different outcomes.
The fundamental counting principle can also be
extended to three or more events. For example,
if three events occur in m, n, and p ways, then
the number of ways that all three events can
occur is m n p
In our example with the die, spinner, and
coin this would result in 6 8 2 or 96
different outcomes.
You are framing a picture. The frames are
available in 12 different styles. Each style
is available in 55 different colors. You also
want a blue mat board which is available
in 11 different shades of blue. How many
different ways can you frame the picture?
The standard configuration for a New
Jersey license plate is one letter, followed
by two digits, followed by 3 letters.
How many different license plates are
possible if letters and digits can be
repeated?
What if letters and digits can not be
repeated?
An ordering of objects is called a
permutation.
How many different ways can you order the red, blue
and yellow cubes from the game of little pig?
R, B, Y
R, Y, B
B, R, Y
B, Y, R
Y, R, B
Y, B, R
An ordering of objects is called a
permutation.
Another way of looking at it…
Area Model:
B, R, Y Y, R, B
R, B, Y
R, Y, B B, Y, R
Y, B, R
An ordering of objects is called a
permutation.
You can use the fundamental counting principle to find
the number of permutations of Red, Blue, and Yellow.
Think of how many choices there are for the first color.
Once the first color has been chosen, how many
choices are there for the second color…
3 2 1 6
An ordering of objects is called a
permutation.
3 2 1 6
The expression representing the number of
permutations in the game of little pig can be written as:
3!
Where the ! symbol is the factorial symbol and the
expression is read “3 factorial.”
If there are ten bands competing in the
battle of the bands next year, in how many
different ways can they finish the
competition?
10! 3,628,800
In how many ways can three of the teams
finish first, second, and third?
10 9 8 720
After your mariachi band wins the battle of
the bands, you decide to burn a demo CD.
Your band has 12 songs stored on your
computer, but you want to put only 4
songs on the demo CD. In how many
orders can you burn 4 of the 12 songs onto
the CD?
12
P4
12 11 10 9 11,880
Find the number of distinguishable
permutations of the letters in MIAMI or
TALLAHASSEE.
5!
2! 2!
11!
3! 2! 2! 2!
Think about it this way:
If you consider the Letters E, Y, and E to be
distinct, then there are 6 permutations of E Y E.
EYE
EYE
YEE
YEE
EEY
EEY
However, if you consider the E to be
interchangeable, then there are only three:
EYE
YEE
EEY
Combinations
When playing poker, is the order
in which the cards in your hand
are dealt to you important?
This is where combinations differ from permutations.
As a permutation is 2,8,7,5 different from 8,5,7,2?
These would be the same combination.
Combinations
If the order in which the cards are dealt is not
important, how many different 5 card hands are
possible?
52
C5
52 51 50 49 48
5!
Combinations
Consider a simple example of 5
cubes. A red, yellow, blue, green,
and purple.
How many combinations of three are there?
RYBGP
5
RYB
RYG
C3
RYP
RBG
5 4 3
3!
RBP
RGP
BGP
YBG
YBP
YGP
Combinations
In how many five card hands are all 5 cards red?
26
C5
26 25 24 23 22
5!
65780
In how many five card hands are all 5 cards the
same color?
65780 2
131,560
William Shakespeare wrote 38
plays that can be divided into three
genres. Of the 38 plays, 18 are
comedies, 10 are histories, and 10
are tragedies.
How many different combinations of 2 comedies
and one tragedy can you read?
18 17
2!
10
1!
153
10
1530
William Shakespeare wrote 38
plays that can be divided into three
genres. Of the 38 plays, 18 are
comedies, 10 are histories, and 10
are tragedies.
How many different sets of at most 3 plays can you
read?
38 37 36
3!
38 37
2!
38
1!
8436
703
38
1
During the school year, the girl’s
basketball team is scheduled to
play 12 home games. You want to
attend at least three of the games.
How many different combinations
of games can you attend.
When considering solutions, should include all
of the orders that we can we a set of games?
What are all of the possible ways to
see games?
12
C0
1
12
C1
12
12
C2
66
12
C3
220
12
C4
495
12
C5
792
12
C6
924
12
C7
792
12
C8
495
12
C9
220
12
C10
66
12
C11
12
12
There are 4096 possibilities for attending games
There are 4017 possibilities for attending at least
3 games.
C12
1
What are all of the possible
outcomes?
Remember, you have a choice to
attend or not attend every game.
If you consider the first two games how many possible
outcomes are there:
A
0
2 2 2
A
A,A
0,A
0
A,0
0,0
What are all of the possible outcomes?
4096 (1 12 66) 4017
12
2
4096
Theoretical Probability Review
P(flipping tails) .5
1
P(rolling a five) 6
These are examples of Theoretical Probability
the number of outcomes you're interested in
the total number of possible outcomes
P(rolling a 1 or 2)
2
6
1
3
Using Permutations or Combinations
to Calculate Probability
One evening 7 of the bands in the
Battle of the Bands are scheduled
to perform. The order in which
they perform is randomly selected.
What is the probability that the bands are selected
in alphabetical order?
1
7!
Using Permutations or Combinations
to Calculate Probability
One evening 7 of the bands in the
Battle of the Bands are scheduled
to perform. The order in which
they perform is randomly selected.
You know the members of four of the bands.
What is the probability that you know the first two
bands to perform?
C2
7 C2
4
The Lottery
Find the probability of a pick 6, where you can choose
6 of 48 numbers and the order is not important.
1
48 C6
Find the probability of selecting four numbers, each
must be an integer from 0 to 9, and the order matters.
1
10 9 8 7
Challenge
You are playing mastermind
with only four colors.
RYBG
What is the probability that you guess all four
pegs correctly on the first try? All four pegs have
to be in the correct order.
What is the probability that you have exactly one
peg correct on the first guess.
If you are told that two of your pegs are correct,
and you then switch two, what is the probability
that all four pegs are now correct?
Probabilities of Mutually Exclusive Events
Students on the Soccer Team
Students on the Debate Team
In a class of 12 students, 7 of the students are on the
soccer team, and 5 are on the debate team.
What is the probability of randomly selecting a student
from the class that is on the soccer team?
What is the probability of randomly selecting a student
from the class that is on the soccer team or debate
team? How about the soccer team AND debate?
Probabilities of Overlapping Events
Students on the Soccer Team
Students on the Debate Team
In a class of 10 students, 7 of the students are on the
soccer team, and 5 are on the debate team.
What is the probability of randomly selecting a student
from the class that is on the debate team?
What is the probability of randomly selecting a student
from the class that is on the soccer team or debate
team? How about the soccer team AND debate?
The Union or Intersection of two events is called a
Compound Event
A or B
A and B
These are overlapping
13
P( A or B)
1
13
4
P( A and B)
13
This is mutually
exclusive
9
P( A or B)
1
9
P( A and B) 0
A card is randomly selected from a
standard deck of 52. What is the
probability that it is a 10 or a face card?
P( A or B)
P( A) P( B)
P( A or B)
4 12
52 52
P( A or B)
16
52
4
13
A card is randomly selected from a
standard deck of 52. What is the
probability that it is a face card or
a heart?
P( A or B) P( A) P(B) P( A and B)
P( A or B)
P( A or B)
13 12 3
52 52 52
22
52
11
26
2 Dice Sums
What is the probability of rolling
5
an 8?
P(8)
36
What is the
probability of not
rolling an 8?
This is called the
complement of P(8).
5
P(not 8) 1
36
31
36
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
The Probability of Independent Events
Two Events are independent when one
has no effect on the other.
Two coin tosses are independent because the
coin has no memory.
Find the probability of flipping:
Heads then Tails
Heads then Heads
Heads, Heads, Tails
Heads, Heads, Heads
Heads, Heads, Heads, Heads
The Probability of Independent Events
Two Events are independent when one
has no effect on the other.
Two coin tosses are independent because the
coin has no memory.
T
Probability of
getting only heads:
1
2
1
4
1
8
1
16
H
H
T
The Probability of Independent Events
Two Events are independent when one
has no effect on the other.
The probability of flipping heads twice is half of
a half, or…
1 1 1
P(H,H)
P(H,H,H)
2
2
1
2
1
2
4
1
2
1
8
So, to find the probability of independent events
occurring, multiply the probabilities.
A spinner is divided into ten equal
regions 1 to 10. What is the
probability that three consecutive
spins will result in perfect squares?
How many possible results are there for spinning
the spinner 3 times? Can you get the same result
twice?
10 10 10
How many possible ways are there to spin perfect
squares three times in a row?
3 3 3
P(3 Consecutive Squares)
3 3 3
10 10 10
27
1000
A fundraiser sells 150 tickets to win
a car, and 200 tickets to win a
MacBook. You buy 5 tickets of each.
What is the probability that you will win both?
P(Car and MacBook )
5
5
150 200
25
1
30000 1200
What is the probability that you will win the car, but
not the MacBook?
P(Car and not MacBook )
5 195
150 200
13
400
Sometimes one event affects the
probability of another happening.
Remember the carrier payment
plan. There was a $10 and five $1 in
the paper bag.
What is the probability that the first bill is a $1 and
the second is a $10.
P( $1 then a $10 )
5
6
1
5
5
30
1
6
In the text, these are called dependent events…
P( A and B)
P(A) P(B | A)
Remember, there are still 30
possible outcomes for this game.
$10
$10
P( $1 then a $10 ) $1A $11
5 1
$1B $11
30 6
$1A
$1B
$1C
$1D
$1E
$11
$11
$11
$11
$11
$2
$2
$2
$2
$2
$2
$2
$2
$2
$2
$1C
$11
$2
$2
$1D
$11
$2
$2
$2
$1E
$11
$2
$2
$2
$2
$2
You randomly select two cards
form a standard deck of 52 cards.
What is the probability that the
first card is not a heart and the
second card is.
In this example, you replace the first card before
selecting the second card.
P(A and B)
39 13
52 52
3
16
What if you do NOT replace the first card before
selecting the second.
P(A and B)
39 13
52 51
13
68
2 Dice Sums Table
Review
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Statistics
Interpreting Probability Distributions
These are based on theoretical probabilities.
Consider Two Dice Sums…
Sum
2
3
4
5
6
7
8
9
10
11
12
Frequency
1
2
3
4
5
6
5
4
3
2
1
P(Sum)
1
36
2
36
3
36
4
36
5
36
6
36
5
36
4
36
3
36
2
36
1
36
Statistics
You can use the probability distribution to
answer questions about two dice sums.
What is the most likely sum when rolling two six sided dice?
7
1
What is the probability that the sum is at least ten?
6
Statistics
Measures of Central Tendency
Mean
7.52
Median
7.5
Mode
7 and 9
Outcome
2
3
4
5
6
7
8
9
10
11
12
Frequency
1
5
5
13
11
15
11
15
13
6
5
100
Total
2
15
20
65
66
105
88
135
130
66
60
752
Statistics
Measures of Central Tendency
Mean
7.52
Median
7.5
Mode
7 and 9
The following data represents observations
of the number of minutes a student
entered class before the bell rang over a
ten day period. Calculate the Mean,
Median, and Mode.
Day
1
2
3
4
5
6
7
8
9
10
Minutes Early
4
8
12
15
3
2
6
9
8
7
Ordered
2
3
4
6
7
8
8
9
12
15
Mean
7.4
Median 7.5
Mode 8
Measures of Dispersion
What is the range of the following data
set?
Day
1
2
3
4
5
6
7
8
9
10
Minutes Early
4
8
12
15
3
2
6
9
8
7
Ordered
2
3
4
6
7
8
8
9
12
15
Range 15 2
13
Another measure of dispersion is called the
Standard Deviation of a data set.
144.4
10
3.8
Day
Minutes
Early
1
2
3
4
5
6
7
8
9
10
4
8
12
15
3
2
6
9
8
7
Difference
from the
mean
-3.4
0.6
4.6
7.6
-4.4
-5.4
-1.4
1.6
0.6
-0.4
Difference
Squared
11.56
0.36
21.16
57.76
19.36
29.16
1.96
2.56
0.36
0.16
144.4
On the 11th day, you come in an hour early
to take a test.
Find the new
Mean,
Median,
Mode,
Range, and
Standard
Deviation of
the Data Set.
Day
Minutes
Early
1
2
3
4
5
6
7
8
9
10
11
4
8
12
15
3
2
6
9
8
7
60
Difference
from the
mean
Difference
Squared
On the 11th day, you come in an hour early
to take a test.
Day
Minutes
Early
1
2
3
4
5
6
7
8
9
10
11
4
8
12
15
3
2
6
9
8
7
60
Mean 12.18
Median 8
Mode 8
Range 58
2659.64
11
15.55
Difference
from the
mean
-8.18
-4.18
-0.18
2.82
-9.18
-10.18
-6.18
-3.18
-4.18
-5.18
47.82
Difference
Squared
66.91
17.47
0.03
7.95
84.27
103.63
38.19
10.11
17.47
26.83
2286.75