math 105: finite mathematics 6-3: the multiplication...
TRANSCRIPT
Introduction Examples Conclusion
MATH 105: Finite Mathematics6-3: The Multiplication Principle
Prof. Jonathan Duncan
Walla Walla College
Winter Quarter, 2006
Introduction Examples Conclusion
Outline
1 Introduction
2 Examples
3 Conclusion
Introduction Examples Conclusion
Outline
1 Introduction
2 Examples
3 Conclusion
Introduction Examples Conclusion
Counting with Tree Diagrams
In the last section we counted the number of elements incombinations of sets with known sizes. But what if we aren’t toldthe size of a set to begin with?
Example
A value meal consists of your choice of one sandwich and one sidedish from a menu with three sandwiches and four side dishes. Howmany possible meals are there?
S3
s1s2s3s4
S2
s1s2s3s4
S1
s1s2s3s4 If there are the same number of
branches at each level, we cansimply multiply.
3 · 4 = 12
sandwich sides meals
Introduction Examples Conclusion
Counting with Tree Diagrams
In the last section we counted the number of elements incombinations of sets with known sizes. But what if we aren’t toldthe size of a set to begin with?
Example
A value meal consists of your choice of one sandwich and one sidedish from a menu with three sandwiches and four side dishes. Howmany possible meals are there?
S3
s1s2s3s4
S2
s1s2s3s4
S1
s1s2s3s4 If there are the same number of
branches at each level, we cansimply multiply.
3 · 4 = 12
sandwich sides meals
Introduction Examples Conclusion
Counting with Tree Diagrams
In the last section we counted the number of elements incombinations of sets with known sizes. But what if we aren’t toldthe size of a set to begin with?
Example
A value meal consists of your choice of one sandwich and one sidedish from a menu with three sandwiches and four side dishes. Howmany possible meals are there?
S3
s1s2s3s4
S2
s1s2s3s4
S1
s1s2s3s4 If there are the same number of
branches at each level, we cansimply multiply.
3 · 4 = 12
sandwich sides meals
Introduction Examples Conclusion
Multiplication Principle
Multiplication Principle of Counting
If a task consists of a sequence of choices in which there are pselections for the first choice, q selections for the second choice, rselections for the third choice, and so on, then the task of makingthese selections can be done in
p · q · r · . . .
different ways.
Note
The number of selections available for the second choice can notdepend on which first choice is made.
Introduction Examples Conclusion
Multiplication Principle
Multiplication Principle of Counting
If a task consists of a sequence of choices in which there are pselections for the first choice, q selections for the second choice, rselections for the third choice, and so on, then the task of makingthese selections can be done in
p · q · r · . . .
different ways.
Note
The number of selections available for the second choice can notdepend on which first choice is made.
Introduction Examples Conclusion
Outline
1 Introduction
2 Examples
3 Conclusion
Introduction Examples Conclusion
Car Production
Example
A certain type of car can be purchased in any of five colors, with amanual or automatic transmission, and with any of three enginesizes. How many different car packages are available?
Note
It is still possible to draw a tree diagram in this example. It would,however, take more time than multiplying.
Introduction Examples Conclusion
Car Production
Example
A certain type of car can be purchased in any of five colors, with amanual or automatic transmission, and with any of three enginesizes. How many different car packages are available?
5 · 2 · 3 = 12
color transmission engine packages
Note
It is still possible to draw a tree diagram in this example. It would,however, take more time than multiplying.
Introduction Examples Conclusion
Car Production
Example
A certain type of car can be purchased in any of five colors, with amanual or automatic transmission, and with any of three enginesizes. How many different car packages are available?
5 · 2 · 3 = 12
color transmission engine packages
Note
It is still possible to draw a tree diagram in this example. It would,however, take more time than multiplying.
Introduction Examples Conclusion
License Plates
Example
Let L be the set of Washington state license plates–three numbersfollowed by three letters. How many license plates are in the set?
Example
Now suppose that letters and digits a license plate may not berepeated. How many possible plates are there?
Introduction Examples Conclusion
License Plates
Example
Let L be the set of Washington state license plates–three numbersfollowed by three letters. How many license plates are in the set?
L = {AAA 000,AAA 001,AAA 002, . . .}
Example
Now suppose that letters and digits a license plate may not berepeated. How many possible plates are there?
Introduction Examples Conclusion
License Plates
Example
Let L be the set of Washington state license plates–three numbersfollowed by three letters. How many license plates are in the set?
L = {AAA 000,AAA 001,AAA 002, . . .}
c(L) =26 · 26 · 26 · 10 · 10 · 10
= 17,576,000Letters Digits
Example
Now suppose that letters and digits a license plate may not berepeated. How many possible plates are there?
Introduction Examples Conclusion
License Plates
Example
Let L be the set of Washington state license plates–three numbersfollowed by three letters. How many license plates are in the set?
L = {AAA 000,AAA 001,AAA 002, . . .}
c(L) =26 · 26 · 26 · 10 · 10 · 10
= 17,576,000Letters Digits
Example
Now suppose that letters and digits a license plate may not berepeated. How many possible plates are there?
Introduction Examples Conclusion
License Plates
Example
Let L be the set of Washington state license plates–three numbersfollowed by three letters. How many license plates are in the set?
L = {AAA 000,AAA 001,AAA 002, . . .}
c(L) =26 · 26 · 26 · 10 · 10 · 10
= 17,576,000Letters Digits
Example
Now suppose that letters and digits a license plate may not berepeated. How many possible plates are there?
c(L) =26 · 25 · 24 · 10 · 9 · 8
= 11,232,000Letters Digits
Introduction Examples Conclusion
Seating Arrangmenets
Example
There are 8 seats in the front row of a classroom, and 12 eagerstudents wishing to fill them. In how many ways can these seats beassigned?
Note
This type of multiplication principle problem is very typical and willbe given a special designation in the next section.
Introduction Examples Conclusion
Seating Arrangmenets
Example
There are 8 seats in the front row of a classroom, and 12 eagerstudents wishing to fill them. In how many ways can these seats beassigned?
12 · 11 · 10 · 9 · 8 · 7 · 6 · 5 = 19, 958, 400
Note
This type of multiplication principle problem is very typical and willbe given a special designation in the next section.
Introduction Examples Conclusion
Seating Arrangmenets
Example
There are 8 seats in the front row of a classroom, and 12 eagerstudents wishing to fill them. In how many ways can these seats beassigned?
12 · 11 · 10 · 9 · 8 · 7 · 6 · 5 = 19, 958, 400
Note
This type of multiplication principle problem is very typical and willbe given a special designation in the next section.
Introduction Examples Conclusion
Failure of the Multiplication Principle
Example
Recall our automibile which came in 5 colors, with 2 transmissionsand 3 engine sizes. Suppose that the smallest engine only camewith an automatic transmission. How many packages are nowavailable?
Introduction Examples Conclusion
Failure of the Multiplication Principle
Example
Recall our automibile which came in 5 colors, with 2 transmissionsand 3 engine sizes. Suppose that the smallest engine only camewith an automatic transmission. How many packages are nowavailable?
5 · 3 · ? = ?
colors engines transmission packages
Introduction Examples Conclusion
Failure of the Multiplication Principle
Example
Recall our automibile which came in 5 colors, with 2 transmissionsand 3 engine sizes. Suppose that the smallest engine only camewith an automatic transmission. How many packages are nowavailable?
5 · 3 · ? = ?
colors engines transmission packages
Introduction Examples Conclusion
Failure of the Multiplication Principle
Example
Recall our automibile which came in 5 colors, with 2 transmissionsand 3 engine sizes. Suppose that the smallest engine only camewith an automatic transmission. How many packages are nowavailable?
5 · 3 · ? = ?
colors engines transmission packages
Introduction Examples Conclusion
Failure of the Multiplication Principle
Example
Recall our automibile which came in 5 colors, with 2 transmissionsand 3 engine sizes. Suppose that the smallest engine only camewith an automatic transmission. How many packages are nowavailable?
5 · 3 · ? = 25
colors engines transmission packages
Introduction Examples Conclusion
Outline
1 Introduction
2 Examples
3 Conclusion
Introduction Examples Conclusion
Important Concepts
Things to Remember from Section 6-3
1 Tree Diagrams can be very useful!
2 Multiplication Principle
3 There are cases where the Multiplication Principle does notwork!
Introduction Examples Conclusion
Important Concepts
Things to Remember from Section 6-3
1 Tree Diagrams can be very useful!
2 Multiplication Principle
3 There are cases where the Multiplication Principle does notwork!
Introduction Examples Conclusion
Important Concepts
Things to Remember from Section 6-3
1 Tree Diagrams can be very useful!
2 Multiplication Principle
3 There are cases where the Multiplication Principle does notwork!
Introduction Examples Conclusion
Important Concepts
Things to Remember from Section 6-3
1 Tree Diagrams can be very useful!
2 Multiplication Principle
3 There are cases where the Multiplication Principle does notwork!
Introduction Examples Conclusion
Next Time. . .
The multiplication principle is the bases for much of the countingwe will do in this class. It is an important concept to know andpractice.
Next time we will examine a specific type of MultiplicationPrinciple problem which results in a counting rule called a“Permutation”.
For next time
Read Section 6-4 (pp 336-342)
Do Problem Sets 6-3 A,B
Introduction Examples Conclusion
Next Time. . .
The multiplication principle is the bases for much of the countingwe will do in this class. It is an important concept to know andpractice.
Next time we will examine a specific type of MultiplicationPrinciple problem which results in a counting rule called a“Permutation”.
For next time
Read Section 6-4 (pp 336-342)
Do Problem Sets 6-3 A,B