Linear-Circular-and-Repetition-Permutation.pptx

PERMUTATION OF AN OBJECT

1. Permutation

2. Permutation Permutation is an arrangement of objects in a specific order. Permutation also refers to any one of all possible arrangements of the elements of the given set. Permutation is when the order or arrangement is IMPORTANT .

3. Example #1 If there are three students and three vacant chairs in front how many possible arrangements can we arrange the students?

4. Therefore there are 6 possible arrangement. S1-S2-S3 S3-S1-S2 S2-S1-S3 S1-S3-S2 S2-S3-S1 S3-S2-S1

5. !!! Instead of listing all the possible arrangement there is an easy way to know how many possible arrangements are there.

6. The total number of permutations of n objects taken r at a time is given by the expression: P(n,r)= n(n-1)(n-2)….(n-r+1)

7. Example #2: How many different ways can 6 different books be arranged on a shelf?

8. Solution: Given: n= 6 books P(n,r)= n(n-1)(n-2)….(n-r+1) P(n,r)= 6 (6-1)(6-2)(6-3)(6-4)(6-5) P(n,r)= 6 (5)(4)(3)(2)(1) P(n,r)= 720 Simply P(n,r)= 6! = 720

9. Factorial (!) Means to multiply a series of descending natural numbers. It's a shorthand way of writing numbers the product of all positive integers less than or equal to n n n!     1 1 1 1 2 2 ×  1 = 2 ×  1! = 2 3 3 ×  2 × 1 = 3 ×  2! = 6 4 4 ×  3 × 2 × 1 = 4 ×  3! = 24 5 5 ×  4 × 3 × 2 × 1 = 5 ×  4! = 120

10. Example # 3: Suppose you work at a music store and have four CDs you wish to arrange from left to right on a display shelf. The four CDs are hip-hop, country, rock, and alternative (shorthand: H, C, R, A ). How many options do you have?

11. Solution: Given: n= 4 P(n,r)= n(n-1)(n-2)….(n-r+1) P(n,r)= 4 (4-1)(4-2)(4-3) P(n,r)= 4(3)(2)(1) P(n,r)= 24 Therefore , the CD’s can be arranged in 24 different ways.

12. Example #4: A father, mother, 2 boys, and 3 girls are asked to line up for a photograph. Determine the number of ways they can line up if: a. there are no restrictions b. the parents stand together c. all the females stand together

13. Solution: there are no restrictions n= 7! n= 7 x 6 x 5 x 4 x 3 x 2 x 1 n= 5040 Therefore, the family members can be line up in 5040 ways.

14. Solution: the parents stand together n= 6! 2! n= (6x 5 x 4 x 3 x 2 x 1) (2 x 1) n= (720)(2) n= 1440 Therefore, the family members can be line up in 1440 ways if the parents stand together.

15. Solution: all the females stand together n= 4! 4! n= (4 x 3 x 2 x 1)(4 x3 x 2 x 1) n= (24)(24) n= 576 Therefore, there are 576 different ways the family can line up if the females stand together.

16. Example #5: Aaron has 3 math books, 4 science books, and 5 history books. How many different ways can these books be arranged on a shelf if books of the same subject must be kept together?

17. Solution: n= (3!) x (5!)(4!)(3!) n= (6) x (120)(24)(6) n= (6) x (17, 280) n= 103 680

18. Example # 6: The Racing Club organizes a race in which 5 cars A, B, C, D, and E are joined. How many ways can the first two positions be filled if there are no ties?

19. Solution: * we used to represent the number of possible permutations of 5 things when only 2 out of the 5 things are taken.

20. Place your screenshot here the number of permutations that can be made with n things taken r at a time

21. The number of permutations of n things taken r at a time is given by: = n(n-1)(n-2)…(n-r+1) Look at the last factor: n-2 = n- (3-1) = n- (r-1) = n-r+1 NOTE : n ≥ r =n(n-1)(n-2)…(n-r+1)

22. Example # 7: In a school club, there are 5 possible choices for the president, a secretary, a treasurer, and an auditor. Assuming that each of them is qualified for any of these position. In how many ways can the 4 officers be selected?

23. Solution: = n(n-1)(n-2)…(n-r+1) = 5 (5-1)(5-2)(5-3) = 5 x 4 x 3 x 2 = 120 ways = = = = = 120  

24. Example # 8: The Open Minded Band has 20 songs to perform in a concert. At the upcoming Battle of the Bands,they will play 2 songs. In how many different orders can they perform two of their songs?

25. Solution: 20 P 2 = 20 P 2 = 380  

26. The number of permutations of n things taken n at a time is given by: = n(n-1)(n-2)…(3)(2)(1)= n!

27. Example # 9: The Racing Club organizes a race in which 5 cars A, B, C, D, and E are joined. How many possible race results if there are no ties?

28. Solution:

29. Circular Permutation

30. Place your screenshot here Are the permutation of objects when they are arranged in a crcular pattern Is a special case of permutation where the arrangements of things is in a circular pattern.

31. Place your screenshot here Example # 10: In how many ways can 3 people be seated around a circular table?

32. Solution: Given: n= 3 Therefore, 3! 3!= 3 X 2 X 1 3!= 6 A-B-C A-C-B B-C-A B-A-C C-A-B C-B-A

33. Observe that all the arrangements falling on the same column are just the same because the 3 people are supposed to be seated around a circular table. There are 6 arrangements.

34. A C B A C B there are only 2 possible permutation in arranging 3 persons at a round table .

35. Therefore the circular permutations, P of 3 objects is: P= P= P= P= 2  

36. Did you know that, P= (n – 1)!

37. Example # 11: In how many different ways can you arranged 8 figures on a circular shelf?

38. Solution: Given: n= 8 P= (n-1)! P= (8-1)! P= 7! P= 5, 040 P= P= P= P= 5 040  

39. Example # 12: Mr. Kesler is at a picnic that has a circular revolving condiment server. Six condiments are placed on the table. How many ways can the condiments be arranged?

40. Solution: Given: n= 6 P= (n-1)! P= (6-1)! P= 5! P= 120 P= P= P= P= 120  

41. Example # 13: Eleven people are to be seated at a round table where one person is seated closest to the exit. How many possible arrangements of people relative to the exit are possible?

42. Solution: Given: n= 11 P= (n-1)! P= (11-1)! P= 10! P= 3, 628, 800 P= P= P= P= 3, 628, 800  

43. Permutation with REPETITION

44. Example # 14: How many distinguishable permutation are there for the letters of the word TENNESSEE taken all together?

45. Solution: Given: Letters in all= 9 E’s= 4 N’s= 2 S’s= 2 P= P= P= P= 3780  

46. Example # 15: Find the number of distinguishable permutations of the letters in the word “ BASKETBALL ”.

47. Solution: Given: All letters: 10 B’s= 2 A’s = 2 L’s= 2 P= P= 453, 600 ways  

48. THANK YOU! 

49. SlidesCarnival icons are editable shapes. This means that you can: Resize them without losing quality. Change fill color and opacity. Change line color, width and style. Isn’t that nice? :) Examples:

50. Now you can use any emoji as an icon! And of course it resizes without losing quality and you can change the color. How? Follow Google instructions https://twitter.com/googledocs/status/730087240156643328 ✋👆👉👍👤👦👧👨👩👪💃🏃💑❤😂😉😋😒😭👶😸🐟🍒🍔💣📌📖🔨🎃🎈🎨🏈🏰🌏🔌🔑 and many more... 😉