chapter 11 – counting methods intro to counting methods section 11.1: counting by systematic...
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Chapter 11 – Counting MethodsIntro to Counting Methods
Section 11.1: Counting by Systematic Listing
There are 3 ways to systematically list all the possible outcomes of an event:
1. Listing all Possiblities
2. Product Tables
3. Tree Diagrams
How many ways can a 6 member club elect a President?
Club Members: Aaron, Bob, Carly, Debbie, Eddie and Frank
Six Possible Presidents: Aaron, Bob, Carly, Debbie, Eddie, Frank
How many ways can a 6 member club elect a President and Vice President?
Systematic Listing of Club Members: Aaron, Bob, Carly, Debbie, Eddie, FrankFor simplicity lets let: A =Aaron, B=Bob, C =Carly, D =Debbie, E =Eddie, F =Frank
We will list the President first and the Vice President second.
Choose someone to be President.
A
Then list all the possible Vice Presidents with your President.
We will list the President first and the Vice President second.
Choose someone to be President.
AB
Then list all the possible Vice Presidents with your President.
ACADAEAF
Choose another President and match all the possible Vice Presents with that President.
ABACADAEAF
BABCBDBEBF
Continue this process until you have listed all possible President-Vice President combinations.
ABACADAEAF
BABCBDBEBF
CACBCDCECF
DADBDCDEDF
EAEBECEDEF
FAFBFCFDFE
Is the combination AB the same as BA? Why or why not?
Try this one on your own…
Consider two six sided cubes. The first one is numbered 1 – 6 and the second one is lettered A – F. What are all the possible outcomes of rolling these two cubes together?
1A 2A 3A 4A 5A 6A1B 2B 3B 4B 5B 6B1C 2C 3C 4C 5C 6C1D 2D 3D 4D 5D 6D1E 2E 3E 4E 5E 6E1F 2F 3F 4F 5F 6F
Why does this event have 36 possible outcomes while Example 1 only has 30 outcomes?
2. Product Tables
When we have an event that involves two tasks (i.e. Choosing a President and a Vice President or rolling two cubes) we can use a product table to show all possible combinations.
Listing the presidents across the top and vice presidents along the side we have….
ABACADAE AF
BA
BCBDBE BF
CACB
CDCE CF
DADBDC
DE DF
EAEBECED
EF
Vic
e P
resi
dent
President
A B C D EF
A B C D E F FAFBFCFDFE
Construct a product table to show all the possible outcomes for our numbered and lettered cubes problem.
Lette
r C
ube
Number Cube
A B C D EF
1 2 3 4 5 61A 2A 3A 4A 5A 6A1B 2B 3B 4B 5B 6B1C 2C 3C 4C 5C 6C1D 2D 3D 4D 5D 6D1E 2E 3E 4E 5E 6E1F 2F 3F 4F 5F 6F
3. Tree Diagrams
Tree diagrams are another way to systematically show all possible outcomes. Tree diagrams are useful for events containing three or more parts.
Here is a tree diagram of our President-Vice Presidentproblem.
Vice PresidentPresident
AB D EF
A
AB C D E
AB C D F
ABC EF
A C D EF
B C D EF
B
C
D
E
F
Construct a tree diagram to show all the possible outcomes for our numbered and lettered cubes problem.
Letter CubeNumber Cube
1
AB C D EF
2
3
4
5
6
AB C D EF
AB C D EF
AB C D EFAB C D EF
AB C D EF
Example 7a: Michelle’s computer printer allows for optional settings with a panel of four on-off switches. Construct a tree diagram to show all the possible ways the switches can be set.
Switch #1 Switch #2 Switch #3 Switch #4On Off
On Off
On Off
On Off
On Off
On Off
On Off
On Off
On
Off
On
Off
On
Off
On
Off
On
Off
On
Off
On
Off
On On On On
On On On
On On On
On On
On On On
On On
On On
On
On
Off
Off
Off Off
Off
Off Off
Off Off
Off Off Off
Off
Off Off
Off Off
Off Off Off
Off Off
Off Off Off
Off O
On On
On
ff Off
On
On On
On
On On
On
O
Off Off
n
Off Off
Notice if we let ON be represented by 0, and OFF represented by 1
1
1
1 1
1
1 1
1 1
1 1 1
1
0 0 0 0
0 0 0
0 0 0
0 0
0 0 0
0 0
0 0
0
0 0 0
0 0
0 0
0
0 0
1 1
1 1
1 1 1
1 1
1 1 0
0
1
1 1 1
1 1 1 1
Example 7b: Michelle’s computer printer allows for optional settings with a panel of four on-off switches. Construct a tree diagram to show all the possible ways the switches can be set if no two adjacent switches can both be off.
Switch #1 Switch #2 Switch #3 Switch #4On Off
On Off
On Off
On Off
On Off
On Off
On Off
On Off
On
Off
On
Off
On
Off
On
Off
On
Off
On
Off
On
Off
Switch #1 Switch #2 Switch #3 Switch #4On Off
On
On Off
On Off
On
On
Off
On
On
Off
On
Off
On
On
Off
On On On On
On On On
On On On
On On
On On On
On On
On On
On
On
Off
Off
Off Off
Off
Off Off
Off Off
Off Off Off
Off
Off Off
Off Off
Off Off Off
Off Off
Off Off Off
Off O
On On
On
ff Off
On
On On
On
On On
On
O
Off Off
n
Off Off
Notice if we let ON be represented by 0, and OFF represented by 1
1
1
1 1
1
1 1
1 1
1 1 1
1
0 0 0 0
0 0 0
0 0 0
0 0
0 0 0
0 0
0 0
0
0 0 0
0 0
0 0
0
0 0
1 1
1 1
1 1 1
1 1
1 1 0
0
1
1 1 1
1 1 1 1
Example 8: Aaron, Bobby, Chuck and Debbie have tickets for four reserved seats in a row at a concert. What are the different ways they can seat themselves so that Aaron and Bobby will sit next to each other? Construct a tree diagram.
First Second Third Fourth
Seat Seat Seat SeatC
D
C
D
B
A A B
B
A
A B
B
A
A
B
D
A
B
C
A
B
C
D
D
C
D
C
D
D B A
C
C
B A
A B C D
A B D C
B A C D
B A D C
C A B D
C B A D
C D A B
C D B A
D A B C
D B A C
D C A C
D C B A