12x1 t09 07 arrangements in a circle
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Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placing objects in a circle?
Permutations
Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placing objects in a circle?The difference is the number of ways the first object can be placed.
Permutations
Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placing objects in a circle?The difference is the number of ways the first object can be placed.
Line
Permutations
Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placing objects in a circle?The difference is the number of ways the first object can be placed.
Line
Permutations
Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placing objects in a circle?The difference is the number of ways the first object can be placed.
Line
In a line there is a definite start and finish of the line.
Permutations
Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placing objects in a circle?The difference is the number of ways the first object can be placed.
Line
In a line there is a definite start and finish of the line.
The first object has a choice of 6 positions
Permutations
Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placing objects in a circle?The difference is the number of ways the first object can be placed.
Line
In a line there is a definite start and finish of the line.
The first object has a choice of 6 positions
Circle
Permutations
Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placing objects in a circle?The difference is the number of ways the first object can be placed.
Line
In a line there is a definite start and finish of the line.
The first object has a choice of 6 positions
Circle
Permutations
Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placing objects in a circle?The difference is the number of ways the first object can be placed.
Line
In a line there is a definite start and finish of the line.
The first object has a choice of 6 positions
CircleIn a circle there is no definite start or finish of the circle.
Permutations
Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placing objects in a circle?The difference is the number of ways the first object can be placed.
Line
In a line there is a definite start and finish of the line.
The first object has a choice of 6 positions
CircleIn a circle there is no definite start or finish of the circle.
It is not until the first object chooses its position that positions are defined.
Permutations
Line
Circle
Line
ntsArrangemen ofNumber 1object for
iespossibilit
1tsArrangemen ofNumber 1object for
iespossibilit
Circle
Line
ntsArrangemen ofNumber 1object for
iespossibilit
1 n
2object for iespossibilit
Circle
1tsArrangemen ofNumber 1object for
iespossibilit
1 n
2object for iespossibilit
Line
ntsArrangemen ofNumber 1object for
iespossibilit
1 n 2 n
2object for iespossibilit
3object for iespossibilit
Circle
1tsArrangemen ofNumber 1object for
iespossibilit
1 n 2 n
2object for iespossibilit
3object for iespossibilit
Line
ntsArrangemen ofNumber 1object for
iespossibilit
1 n 2 n 1
2object for iespossibilit
3object for iespossibilit
objectlast for iespossibilit
Circle
1tsArrangemen ofNumber 1object for
iespossibilit
1 n 2 n 1
2object for iespossibilit
3object for iespossibilit
objectlast for iespossibilit
Line
Circle
ntsArrangemen ofNumber 1object for
iespossibilit
1 n 2 n 1
2object for iespossibilit
3object for iespossibilit
objectlast for iespossibilit
1tsArrangemen ofNumber 1object for
iespossibilit
1 n 2 n 1
2object for iespossibilit
3object for iespossibilit
objectlast for iespossibilit
nn! circle ain tsArrangemen ofNumber
Line
Circle
ntsArrangemen ofNumber 1object for
iespossibilit
1 n 2 n 1
2object for iespossibilit
3object for iespossibilit
objectlast for iespossibilit
1tsArrangemen ofNumber 1object for
iespossibilit
1 n 2 n 1
2object for iespossibilit
3object for iespossibilit
objectlast for iespossibilit
nn! circle ain tsArrangemen ofNumber
!1 n
e.g. A meeting room contains a round table surrounded by ten chairs.
(i) A committee of ten people includes three teenagers. How many arrangements are there in which all three sit together?
e.g. A meeting room contains a round table surrounded by ten chairs.
(i) A committee of ten people includes three teenagers. How many arrangements are there in which all three sit together?
!3tsArrangemen
arrangedbecan teenagersthreethe waysofnumber the
e.g. A meeting room contains a round table surrounded by ten chairs.
(i) A committee of ten people includes three teenagers. How many arrangements are there in which all three sit together?
!3tsArrangemen
arrangedbecan teenagersthreethe waysofnumber the
!7others 7 ) teenagers(3
circle ain objects 8 arrangingofwaysofnumber
e.g. A meeting room contains a round table surrounded by ten chairs.
(i) A committee of ten people includes three teenagers. How many arrangements are there in which all three sit together?
!3tsArrangemen
arrangedbecan teenagersthreethe waysofnumber the
!7others 7 ) teenagers(3
circle ain objects 8 arrangingofwaysofnumber
30240
(ii) Elections are held for Chairperson and Secretary. What is the probability that they are seated directly opposite each other?
(ii) Elections are held for Chairperson and Secretary. What is the probability that they are seated directly opposite each other?
!9ns)restrictio (noWays
(ii) Elections are held for Chairperson and Secretary. What is the probability that they are seated directly opposite each other?
!9ns)restrictio (noWays
1ons)(restrictiWays
circle in the1st are they as
anywheresit canPresident
(ii) Elections are held for Chairperson and Secretary. What is the probability that they are seated directly opposite each other?
!9ns)restrictio (noWays
1ons)(restrictiWays
circle in the1st are they as
anywheresit canPresident
1
Presidentoppositesit
mustSecretary
(ii) Elections are held for Chairperson and Secretary. What is the probability that they are seated directly opposite each other?
!9ns)restrictio (noWays
1ons)(restrictiWays
circle in the1st are they as
anywheresit canPresident
1 !8
Presidentoppositesit
mustSecretary
gocan peopleremainingWays
(ii) Elections are held for Chairperson and Secretary. What is the probability that they are seated directly opposite each other?
!9ns)restrictio (noWays
1ons)(restrictiWays
circle in the1st are they as
anywheresit canPresident
1 !8
Presidentoppositesit
mustSecretary
gocan peopleremainingWays
!9
!811opposite S & P P
(ii) Elections are held for Chairperson and Secretary. What is the probability that they are seated directly opposite each other?
!9ns)restrictio (noWays
1ons)(restrictiWays
circle in the1st are they as
anywheresit canPresident
1 !8
Presidentoppositesit
mustSecretary
gocan peopleremainingWays
!9
!811opposite S & P P
91
(ii) Elections are held for Chairperson and Secretary. What is the probability that they are seated directly opposite each other?
!9ns)restrictio (noWays
1ons)(restrictiWays
circle in the1st are they as
anywheresit canPresident
1 !8
Presidentoppositesit
mustSecretary
gocan peopleremainingWays
!9
!811opposite S & P P
91
Note: of 9 seats only 1 is opposite the
President
91
oppositeP
Sometimes simple logic is quicker!!!!
Seven people are to be seated at a round table2002 Extension 1 HSC Q3a)
(i) How many seating arrangements are possible?
Seven people are to be seated at a round table2002 Extension 1 HSC Q3a)
(i) How many seating arrangements are possible?
!6tsArrangemen
Seven people are to be seated at a round table2002 Extension 1 HSC Q3a)
(i) How many seating arrangements are possible?
!6tsArrangemen 720
Seven people are to be seated at a round table2002 Extension 1 HSC Q3a)
(i) How many seating arrangements are possible?
(ii) Two people, Kevin and Jill, refuse to sit next to each other. How many seating arrangements are then possible?
!6tsArrangemen 720
Seven people are to be seated at a round table2002 Extension 1 HSC Q3a)
(i) How many seating arrangements are possible?
(ii) Two people, Kevin and Jill, refuse to sit next to each other. How many seating arrangements are then possible?
!6tsArrangemen 720
Note: it is easier to work out the number of ways Kevin and Jill are together and subtract from total number of arrangements.
Seven people are to be seated at a round table2002 Extension 1 HSC Q3a)
(i) How many seating arrangements are possible?
(ii) Two people, Kevin and Jill, refuse to sit next to each other. How many seating arrangements are then possible?
!6tsArrangemen 720
Note: it is easier to work out the number of ways Kevin and Jill are together and subtract from total number of arrangements.
!2 tsArrangemen
togetherare Jill &Kevin waysofnumber the
Seven people are to be seated at a round table2002 Extension 1 HSC Q3a)
(i) How many seating arrangements are possible?
(ii) Two people, Kevin and Jill, refuse to sit next to each other. How many seating arrangements are then possible?
!6tsArrangemen 720
Note: it is easier to work out the number of ways Kevin and Jill are together and subtract from total number of arrangements.
!2 tsArrangemen
togetherare Jill &Kevin waysofnumber the
others 5 Jill) &(Kevin circle ain objects 6
arrangingofwaysofnumber
!5
Seven people are to be seated at a round table2002 Extension 1 HSC Q3a)
(i) How many seating arrangements are possible?
(ii) Two people, Kevin and Jill, refuse to sit next to each other. How many seating arrangements are then possible?
!6tsArrangemen 720
Note: it is easier to work out the number of ways Kevin and Jill are together and subtract from total number of arrangements.
!2 tsArrangemen
togetherare Jill &Kevin waysofnumber the
others 5 Jill) &(Kevin circle ain objects 6
arrangingofwaysofnumber
240!5
Seven people are to be seated at a round table2002 Extension 1 HSC Q3a)
(i) How many seating arrangements are possible?
(ii) Two people, Kevin and Jill, refuse to sit next to each other. How many seating arrangements are then possible?
!6tsArrangemen 720
Note: it is easier to work out the number of ways Kevin and Jill are together and subtract from total number of arrangements.
!2 tsArrangemen
togetherare Jill &Kevin waysofnumber the
others 5 Jill) &(Kevin circle ain objects 6
arrangingofwaysofnumber
240!5
240720tsArrangemen 480
Seven people are to be seated at a round table2002 Extension 1 HSC Q3a)
(i) How many seating arrangements are possible?
(ii) Two people, Kevin and Jill, refuse to sit next to each other. How many seating arrangements are then possible?
!6tsArrangemen 720
Note: it is easier to work out the number of ways Kevin and Jill are together and subtract from total number of arrangements.
!2 tsArrangemen
togetherare Jill &Kevin waysofnumber the
others 5 Jill) &(Kevin circle ain objects 6
arrangingofwaysofnumber
240!5
240720tsArrangemen 480
A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code.
1995 Extension 1 HSC Q3a)
(i) How many different codes are possible if letters can be repeated and their order is important?
A B C D E F G H
A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code.
1995 Extension 1 HSC Q3a)
(i) How many different codes are possible if letters can be repeated and their order is important?
With replacementUse Basic Counting Principle
A B C D E F G H
A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code.
1995 Extension 1 HSC Q3a)
(i) How many different codes are possible if letters can be repeated and their order is important?
888 Codes With replacementUse Basic Counting Principle
A B C D E F G H
A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code.
1995 Extension 1 HSC Q3a)
(i) How many different codes are possible if letters can be repeated and their order is important?
888 Codes With replacementUse Basic Counting Principle 512
A B C D E F G H
A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code.
1995 Extension 1 HSC Q3a)
(i) How many different codes are possible if letters can be repeated and their order is important?
888 Codes With replacementUse Basic Counting Principle 512
A B C D E F G H
(ii) How many different codes are possible if letters cannot be repeated and their order is important?
A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code.
1995 Extension 1 HSC Q3a)
(i) How many different codes are possible if letters can be repeated and their order is important?
888 Codes With replacementUse Basic Counting Principle 512
Without replacementOrder is important
Permutation
A B C D E F G H
(ii) How many different codes are possible if letters cannot be repeated and their order is important?
A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code.
1995 Extension 1 HSC Q3a)
(i) How many different codes are possible if letters can be repeated and their order is important?
888 Codes With replacementUse Basic Counting Principle 512
38 Codes P Without replacement
Order is importantPermutation
A B C D E F G H
(ii) How many different codes are possible if letters cannot be repeated and their order is important?
A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code.
1995 Extension 1 HSC Q3a)
(i) How many different codes are possible if letters can be repeated and their order is important?
888 Codes With replacementUse Basic Counting Principle 512
38 Codes P336
Without replacementOrder is important
Permutation
A B C D E F G H
(ii) How many different codes are possible if letters cannot be repeated and their order is important?
(iii) Now suppose that the lock operates by holding 3 buttons down together, so that the order is NOT important.How many different codes are possible?
(iii) Now suppose that the lock operates by holding 3 buttons down together, so that the order is NOT important.How many different codes are possible?
Without replacementOrder is not important
Combination
(iii) Now suppose that the lock operates by holding 3 buttons down together, so that the order is NOT important.How many different codes are possible?
38 Codes C Without replacement
Order is not importantCombination
(iii) Now suppose that the lock operates by holding 3 buttons down together, so that the order is NOT important.How many different codes are possible?
38 Codes C56
Without replacementOrder is not important
Combination
(iii) Now suppose that the lock operates by holding 3 buttons down together, so that the order is NOT important.How many different codes are possible?
38 Codes C56
Without replacementOrder is not important
Combination
Exercise 10I; odds
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