# 4-3 Counting Techniques

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```					4.3 Counting
Techniques
Prob & Stats
Tree Diagrams

   When calculating probabilities,
you need to know the total
outcomes
number of _____________ in
the ______________.
sample space
Tree Diagrams Example
   Use a TREE DIAGRAM to list the
sample space of 2 coin flips. Sample
Space

H
H
Now you could get…
If you got H
T

On the first flip you
could get…..
YOU

H
you could
NowIf you got Tget…
T

T
Tree Diagram Example
   Mr. Arnold’s Closet

3 Shirts            2 Pants

2 Pairs of Shoes
Dress Mr. Arnold
   List all of Mr. Arnold’s outfits   1

2
Dress Mr. Arnold
   List all of Mr. Arnold’s outfits   1

2
3

4
Dress Mr. Arnold
   List all of Mr. Arnold’s outfits       1

2
3

4

5

6
Dress Mr. Arnold
   List all of Mr. Arnold’s outfits       1

2
3

4

5

6
7

8
Dress Mr. Arnold
   List all of Mr. Arnold’s outfits       1

2
3

4

5

6
7

8

9
10
Dress Mr. Arnold
   List all of Mr. Arnold’s outfits       1

2
3

4

5

6
7

8

9
10

11

12
Dress Mr. Arnold                                            If Mr. Arnold picks an
outfit with his eyes
   List all of Mr. Arnold’s outfits       1                           closed…….

2                P(brown shoe) =
3
6/12
1/2
4
P(polo) =
5

6
1/3
4/12
7
P(lookin’ cool) =

8

9
10

11
1
12
Multiplication Rule of Counting

   The size of the sample space is
denominator
the ___________ of our
probability
   So we don’t always need to
know what each outcome is, just
the number of outcomes.
Multiplication Rule of
Compound Events
If…
 X = total number of outcomes
for event A
 Y = total number of outcomes
for event B
 Then number of outcomes for A
followed by B = x times y
_________
Multiplication Rule:
Dress Mr. Arnold

   Mr. Reed had 3 EVENTS

2
shoes
2
pants
3
shirts

How many outcomes are there
for = 12 OUTFITS
2(2)(3)EACH EVENT?
Permutations

   Sometimes we are concerned
with how many ways a group of
arranged
objects can be __________.
•How many ways to arrange books on a
shelf

•How many ways a group of people can
stand
in line

•How many ways to scramble a word’s
letters
   Wonder Woman’s invisible plane has 3
Example:                             
chairs.
There are 3 people who need a lift.
3 People, 3 Chairs                      How many seating options are there?
Think of each chair as
Wonder Woman driving     6 Seating Options!
an EVENT
Batman driving

3          2        1
Now that ways is filled?
How manythe 1st could the
Now the first 2 are filled.
How = options for
many 6 OPTIONS
3(2)(1)many ways to fill ndrd?
st
How 1 chair be filled? 2 3 ?

Superman driving
Example:                  

The batmobile has 5 chairs.
There are 5 people who need a lift.
5 People, 5 Chairs           How many seating options are there?

5     4       3         2       1         =120
Seating Options

Multiply!!
This is a PERMUTATION of 5 objects
Commercial Break:
FACTORIAL

   denoted with !  5!
   Multiply all integers ≤ the number 
5! = 5(4)(3)(2)(1) = 120
   0! = 1
   1! = 1
   Calculate 6!
6! = 6(5)(4)(3)(2)(1) = 720
   What is 6! / 5!?
Commercial Break:
FACTORIAL

   denoted with !  5!
   Multiply all integers ≤ the number 

   0! = 1
   1! = 1
   Calculate 6!

   What is 6! / 5!?
6(5)(4)(3)(2)(1)
5(4)(3)(2)(1)
=6
Example:                  

The batmobile has 5 chairs.
There are 5 people who need a lift.
5 People, 5 Chairs           How many seating options are there?

5!
5     4       3         2       1         =120
Seating Options

Multiply!!
This is a PERMUTATION of 5 objects
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
2nd, and 3rd base?
What if I
choose these
3?

You have to choose 3 AND             Think of the
possibilities!
arrange them
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
2nd, and 3rd base?
What if I
choose these
3?

You have to choose 3 AND             Think of the
possibilities!
arrange them
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
2nd, and 3rd base?
What if I
choose these
3?

You have to choose 3 AND             Think of the
possibilities!
arrange them
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
2nd, and 3rd base?                                 BUT…
What if I
choose
THESE 3?

You have to choose 3 AND            Think of the
possibilities!
arrange them
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
2nd, and 3rd base?                                 BUT…
What if I
choose
THESE 3?

You have to choose 3 AND            Think of the
possibilities!
arrange them
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
2nd, and 3rd base?                                 BUT…
What if I
choose
THESE 3?

You have to choose 3 AND            Think of the
possibilities!
arrange them
Permutations:
Not everyone gets a seat!
   It’s time for annual Justice League softball game.
   How many ways could your assign people to play 1st,
2nd, and 3rd base?                                 BUT…
What if I
choose
THESE 3?

This is going to
take
FOREVER
You have to choose 3 AND            Think of the
possibilities!
arrange them
You have 3 EVENTS?
   How many outcomes for each event

How many
outcomes for this
event!

5

You have to choose 3 AND
arrange them
You have 3 EVENTS?
How many
outcomes for this
event!

4
Now someone is
on FIRST

5

You have to choose 3 AND
arrange them
You have 3 EVENTS?
5(4)(3) = 120 POSSIBLITIES
And on SECOND

4
Now someone is
on FIRST

3                          5
How many
outcomes for this
event!

You have to choose 3 AND
arrange them
Permutation Formula

    You have n objects
    You select r objects
    This is the number of ways you
could select and arrange in
order:

Another common notation for a permutation is nPr
Softball Permutation Revisited
5!
5(4)(3)(2)(1)
5 people to choose from
n=
r = 3 spots to fill (5 – 2!
3)!
2(1)

5(4)(3) = 120 POSSIBLITIES

You have to choose 3 AND
arrange them
Combinations

   Sometimes, we are only
concerned with selecting a group
and not the order in which they
are selected.
   A combination gives the number
of ways to select a sample of r
objects from a group of size n.
Combination: Duty Calls
   There is an evil monster threatening
the city.
   The mayor calls the Justice League.
   He requests that 3 members be sent
to combat the menace.
   The Justice League draws 3 names
out of a hat to decide.
   Does it matter who is selected first?
NOPE
   Does it matter who is selected last?
NOPE
Combination: Duty Calls
Let’s look at the drawing possibilities

STOP!
This is a waste of
time
These are all them
We’ll countthewho gotas
The monster doesn’t care
SAME:
drawn
ONE OUTCOMEfirst.
All these outcomes = same people
We’ll count them as
These are all the SAME:
pounding his face
The monster doesn’t care who got drawn
ONE OUTCOME
first.
All these outcomes = same people
pounding his face
Combination: Duty Calls

Okay, let’s consider other outcomes

10 Possible Outcomes!
Combination Formula

   You have n objects
   You want a group of r objects
   You DON’T CARE what order
they are selected in

Combinations are also denoted nCr
Read “n choose r”
Duty Calls: Revisited
n = 5 people toDOESN’T MATTER
ORDER
choose from
r = 3 spots to fill

5(4)(3)(2)(1)
5!                  20
2
3!(2)!
3!(5 - 3)!
3(2)(1)(2)(1)

10 Possible Outcomes!

Now we can go save the city
Permutation vs. Combination

   Order matters  Permutation
   Order doesn’t matter Combination

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