11X1 T06 05 arrangements in a circle (2010)

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<ul><li> 1. Permutations Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placingobjects in a circle? </li></ul><p> 2. Permutations Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placingobjects in a circle?The difference is the number of ways the first object can be placed. 3. Permutations Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placingobjects in a circle? The difference is the number of ways the first object can be placed.Line 4. Permutations Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placingobjects in a circle? The difference is the number of ways the first object can be placed.Line 5. Permutations Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placingobjects in a circle? The difference is the number of ways the first object can be placed.LineIn a line there is a definite start and finish of the line. 6. Permutations Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placingobjects in a circle? The difference is the number of ways the first object can be placed.LineIn a line there is a definite start and finish of the line. The first object has a choice of 6 positions 7. Permutations Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placingobjects in a circle? The difference is the number of ways the first object can be placed.LineIn a line there is a definite start and finish of the line. The first object has a choice of 6 positionsCircle 8. Permutations Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placingobjects in a circle? The difference is the number of ways the first object can be placed.LineIn a line there is a definite start and finish of the line. The first object has a choice of 6 positionsCircle 9. Permutations Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placingobjects in a circle? The difference is the number of ways the first object can be placed.LineIn a line there is a definite start and finish of the line. The first object has a choice of 6 positionsCircle In a circle there is no definite start or finish of the circle. 10. Permutations Case 4: Ordered Sets of n Objects, Arranged in a CircleWhat is the difference between placing objects in a line and placingobjects in a circle? The difference is the number of ways the first object can be placed.LineIn a line there is a definite start and finish of the line. The first object has a choice of 6 positionsCircle In 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. 11. Line Circle 12. Linepossibilitiesfor object 1 Number of Arrangements nCircle possibilitiesfor object 1 Number of Arrangements 1 13. Line possibilities possibilities for object 2for object 1 Number of Arrangements n n 1Circlepossibilities possibilities for object 2 for object 1 Number of Arrangements 1 n 1 14. Linepossibilities possibilities for object 2 possibilitiesfor object 1for object 3Number of Arrangements n n 1 n 2 Circlepossibilities possibilities for object 2 possibilities for object 1 for object 3Number of Arrangements 1 n 1 n 2 15. Line possibilities possibilities for object 2 possibilities possibilitiesfor object 1for object 3for last object Number of Arrangements n n 1 n 2 1Circlepossibilities possibilities for object 2 possibilities possibilities for object 1 for object 3for last object Number of Arrangements 1 n 1 n 2 1 16. Line possibilities possibilities for object 2 possibilities possibilitiesfor object 1for object 3for last object Number of Arrangements n n 1 n 2 1Circlepossibilities possibilities for object 2 possibilities possibilities for object 1 for object 3for last object Number of Arrangements 1 n 1 n 2 1n! Number of Arrangements in a circle n 17. Line possibilities possibilities for object 2 possibilities possibilitiesfor object 1for object 3for last object Number of Arrangements n n 1 n 2 1Circlepossibilities possibilities for object 2 possibilities possibilities for object 1 for object 3for last object Number of Arrangements 1 n 1 n 2 1n! Number of Arrangements in a circle n n 1! 18. e.g. A meeting room contains a round table surrounded by tenchairs. (i) A committee of ten people includes three teenagers. How many arrangements are there in which all three sit together? 19. e.g. A meeting room contains a round table surrounded by tenchairs. (i) A committee of ten people includes three teenagers. How many arrangements are there in which all three sit together?the number of ways thethree teenagers can be arranged Arrangements 3! 20. e.g. A meeting room contains a round table surrounded by tenchairs. (i) A committee of ten people includes three teenagers. How many arrangements are there in which all three sit together?the number of ways thethree teenagers can be number of ways of arranging arranged 8 objects in a circle(3 teenagers) 7 othersArrangements 3! 7! 21. e.g. A meeting room contains a round table surrounded by tenchairs. (i) A committee of ten people includes three teenagers. How many arrangements are there in which all three sit together?the number of ways thethree teenagers can be number of ways of arranging arranged 8 objects in a circle(3 teenagers) 7 othersArrangements 3! 7! 30240 22. (ii) Elections are held for Chairperson and Secretary.What is the probability that they are seated directly opposite eachother? 23. (ii) Elections are held for Chairperson and Secretary.What is the probability that they are seated directly opposite eachother? Ways (no restrictions) 9! 24. (ii) Elections are held for Chairperson and Secretary.What is the probability that they are seated directly opposite eachother? Ways (no restrictions) 9! President can sit anywhere as they are 1stin the circle Ways (restrictions) 1 25. (ii) Elections are held for Chairperson and Secretary.What is the probability that they are seated directly opposite eachother? Ways (no restrictions) 9! Secretary mustPresident cansit opposite sit anywhere President as they are 1stin the circle Ways (restrictions) 1 1 26. (ii) Elections are held for Chairperson and Secretary.What is the probability that they are seated directly opposite eachother? Ways (no restrictions) 9!Secretary must President can sit opposite sit anywherePresident as they are 1stWays remaining in the circlepeople can goWays (restrictions) 1 1 8! 27. (ii) Elections are held for Chairperson and Secretary.What is the probability that they are seated directly opposite eachother? Ways (no restrictions) 9!Secretary must President can sit opposite sit anywherePresident as they are 1stWays remaining in the circlepeople can goWays (restrictions) 1 1 8!11 8!PP &amp; S opposite 9! 28. (ii) Elections are held for Chairperson and Secretary.What is the probability that they are seated directly opposite eachother? Ways (no restrictions) 9!Secretary must President can sit opposite sit anywherePresident as they are 1stWays remaining in the circlepeople can goWays (restrictions) 1 1 8!11 8!PP &amp; S opposite 9! 1 9 29. (ii) Elections are held for Chairperson and Secretary.What is the probability that they are seated directly opposite eachother? Ways (no restrictions) 9!Secretary must President can sit opposite sit anywherePresident as they are 1stWays remaining in the circlepeople can goNote: of 9 seats onlyWays (restrictions) 1 1 8!1 is opposite thePresident 11 8! Popposite 1PP &amp; S opposite 9! 9 1Sometimes simplelogic is quicker!!!! 9 30. 2002 Extension 1 HSC Q3a) Seven people are to be seated at a round table (i) How many seating arrangements are possible? 31. 2002 Extension 1 HSC Q3a) Seven people are to be seated at a round table (i) How many seating arrangements are possible? Arrangements 6! 32. 2002 Extension 1 HSC Q3a) Seven people are to be seated at a round table (i) How many seating arrangements are possible? Arrangements 6! 720 33. 2002 Extension 1 HSC Q3a) Seven people are to be seated at a round table (i) How many seating arrangements are possible?Arrangements 6! 720(ii) Two people, Kevin and Jill, refuse to sit next to each other. Howmany seating arrangements are then possible? 34. 2002 Extension 1 HSC Q3a) Seven people are to be seated at a round table (i) How many seating arrangements are possible?Arrangements 6! 720(ii) Two people, Kevin and Jill, refuse to sit next to each other. Howmany seating arrangements are then possible? Note: it is easier to work out the number of ways Kevin and Jill are together and subtract from total number of arrangements. 35. 2002 Extension 1 HSC Q3a)Seven people are to be seated at a round table(i) How many seating arrangements are possible? Arrangements 6! 720 (ii) Two people, Kevin and Jill, refuse to sit next to each other. How many seating arrangements are then possible? Note: it is easier to work out the number of ways Kevin and Jill are together and subtract from total number of arrangements. the number of ways Kevin &amp; Jill are together Arrangements 2! 36. 2002 Extension 1 HSC Q3a)Seven people are to be seated at a round table(i) How many seating arrangements are possible? Arrangements 6! 720 (ii) Two people, Kevin and Jill, refuse to sit next to each other. How many seating arrangements are then possible? Note: it is easier to work out the number of ways Kevin and Jill are together and subtract from total number of arrangements. the number of ways number of ways of arranging Kevin &amp; Jill are together6 objects in a circle(Kevin &amp; Jill) 5 others Arrangements 2! 5! 37. 2002 Extension 1 HSC Q3a)Seven people are to be seated at a round table(i) How many seating arrangements are possible? Arrangements 6! 720 (ii) Two people, Kevin and Jill, refuse to sit next to each other. How many seating arrangements are then possible? Note: it is easier to work out the number of ways Kevin and Jill are together and subtract from total number of arrangements. the number of ways number of ways of arranging Kevin &amp; Jill are together6 objects in a circle(Kevin &amp; Jill) 5 others Arrangements 2! 5! 240 38. 2002 Extension 1 HSC Q3a)Seven people are to be seated at a round table(i) How many seating arrangements are possible? Arrangements 6! 720 (ii) Two people, Kevin and Jill, refuse to sit next to each other. How many seating arrangements are then possible? Note: it is easier to work out the number of ways Kevin and Jill are together and subtract from total number of arrangements. the number of ways number of ways of arranging Kevin &amp; Jill are together6 objects in a circle(Kevin &amp; Jill) 5 others Arrangements 2! 5! 240 Arrangements 720 240 480 39. 2002 Extension 1 HSC Q3a)Seven people are to be seated at a round table(i) How many seating arrangements are possible? Arrangements 6! 720 (ii) Two people, Kevin and Jill, refuse to sit next to each other. How many seating arrangements are then possible? Note: it is easier to work out the number of ways Kevin and Jill are together and subtract from total number of arrangements. the number of ways number of ways of arranging Kevin &amp; Jill are together6 objects in a circle(Kevin &amp; Jill) 5 others Arrangements 2! 5! 240 Arrangements 720 240 480 40. 1995 Extension 1 HSC Q3a)A BC D E FG H A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code. (i) How many different codes are possible if letters can be repeated and their order is important? 41. 1995 Extension 1 HSC Q3a)A BC D E FG H A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code. (i) How many different codes are possible if letters can be repeated and their order is important? With replacementUse Basic Counting Principle 42. 1995 Extension 1 HSC Q3a)ABCD E FG H A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code. (i) How many different codes are possible if letters can be repeated and their order is important?Codes 8 8 8With replacementUse Basic Counting Principle 43. 1995 Extension 1 HSC Q3a)ABCD E FG H A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code. (i) How many different codes are possible if letters can be repeated and their order is important?Codes 8 8 8With replacement 512 Use Basic Counting Principle 44. 1995 Extension 1 HSC Q3a)ABCD EF G H A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code. (i) How many different codes are possible if letters can be repeated and their order is important?Codes 8 8 8With replacement 512 Use Basic Counting Principle(ii) How many different codes are possible if letters cannot berepeated and their order is important? 45. 1995 Extension 1 HSC Q3a)ABCD EF G H A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code. (i) How many different codes are possible if letters can be repeated and their order is important?Codes 8 8 8With replacement 512 Use Basic Counting Principle(ii) How many different codes are possible if letters cannot berepeated and their order is important? Without replacement Order is importantPermutation 46. 1995 Extension 1 HSC Q3a)ABCD EF G H A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code. (i) How many different codes are possible if letters can be repeated and their order is important?Codes 8 8 8With replacement 512 Use Basic Counting Principle(ii) How many different codes are possible if letters cannot berepeated and their order is important?Codes 8P3 Without replacement Order is importantPermutation 47. 1995 Extension 1 HSC Q3a)ABCD EF G H A security lock has 8 buttons labelled as shown. Each person using the lock is given a 3 letter code. (i) How many different codes are possible if letters can be repeated and their order is important?Codes 8 8 8With replacement 512 Use Basic Counting Principle(ii) How many different codes are possible if letters cannot berepeated and their order is important?Codes 8P3 Without replacement 336 Order is importantPermutation 48. (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? 49. (iii) Now...</p>