FUNDAMENTAL COUNTING PRINCIPLES AND PERMUTATION.pptx

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1. FUNDAMENTAL COUNTING PRINCIPLES

2. FUNDAMENTAL COUNTING PRINCIPLES If the first operation can be performed in n₁ ways and the second operation in n₂ ways, then the entire experiment can be performed in n₁ x n₂ ways.

3. EXAMPLE #1 In the experiment of tossing a coin and rolling a die, how many elements can be made? n₁ (Coin) = 2 n₂ (Die) = 6 2 x 6 = 12 elements

4. HEAD TAIL 1 2 3 4 5 6 1 2 3 4 5 6 (head, 1) (head, 2) (head, 3) (head, 4) (head, 5) (head, 6) (Tail, 1) (Tail, 2) (Tail, 3) (Tail, 4) (Tail, 5) (Tail, 6) There are 12 elements

5. EXAMPLE #2 How many elements are there in the experiment of choosing a color from red, blue and yellow, and tossing a coin? n1 (Coin) = 2 n2 (Color) = 3 2 x 3 = 6 elements

6. Red (R) Blue (B) Yellow (Y) Head (H) HR HB HY Tail (T) TR TB TY There are 6 elements

7. EXAMPLE #3 Suppose you can have pancake, cereal, or sandwich for your breakfast and juice or milk for your drink. How many choices do you have in all for your breakfast? n1 (Food) = 3 n2 (Drinks) = 2 2 x 3 = 6 choices

8. EXAMPLE #4 Daniel is planning to purchase a photo album. It comes in three sizes, small, medium, and large; and the cover comes in hard or soft bound. The pages can be glossy or silk, and the print can be colored or plain black and white. How many choices does he have for the photo album?

9. n1 (Sizes) = 3 n3 (Pages) = 2 n2 (Cover) = 2 n4 (Print) = 2 3 x 2 x 2 x 2 = 24 Choices

10. PERMUTATION

11. PERMUTATION An arrangement of a given set. In the arrangement of n objects there are n operations involved. The first operation involves choosing an item for the first position; the second operation, choosing an item for the second position and so until the nth operation.

12. LINEAR PERMUTATION FORMULA n! = n x (n-1) x … 1

13. EXAMPLE #1 Suppose Debbie, Anna, Roy and Emily will line up to buy lunch in the school canteen. How many ways can the four students be in line?

14. n = 4 n! = 4! 4! = 4 x 3 x 2 x 1 = 24 ways

15. EXAMPLE #2 In how many ways can the letters of the word “FAITH” be arrange? n = 5 n! = 5! 5! = 5 x 4 x 3 x 2 x 1 = 120 ways

16. EXAMPLE #3 Alex, Alvin, Alyssa, Alfred, Alan, and Aljon are to occupy the front seats of the auditorium. In how many ways can they arrange their seats? n = 6 n! = 6! 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 ways

17. CIRCULAR PERMUTATION FORMULA (n - 1)!

18. EXAMPLE #1 A family with 6 members sit in a round table for dinner. How many ways can the member of the family be seated? n = 6 (n – 1)! = (6 - 1)! 5! = 5 x 4 x 3 x 2 x 1 = 120 ways

19. EXAMPLE #2 In how many ways can 7 appetizers be arranged on a circular tray? n = 7 (n – 1)! = (7 - 1)! 6! = 6 x 5 x 4 x 3 x 2 x 1 = 720 ways

20. PERMUTATION WITH REPETITION OF ELEMENTS  

21. EXAMPLE #1 In how many ways can 2 blue balls, 2 red balls, and a yellow ball be arrange? n = 5 n1 = 2 n2 = 2 =  

22. EXAMPLE #2 In how many ways can the letters of the word “ARRIVE” be arrange? n = 6 n1 = 2 =  

23. EXAMPLE #3 In how many ways can the numbers “012314501” be arrange? n = 9 n1 = 2 n2 = 3 =  

24. PERMUTATION OF N OBJECTS TAKEN R AT A TIME  

25. EXAMPLE #1 How many two digits number can you make from numbers 1, 2, 3, and 4? n = 4 r = 2 4 P 2 = = = 12 ways  

26. EXAMPLE #2 In how many ways can you arrange 6 books in a bookshelf, if the capacity of the bookshelf is only 4 books? n = 6 r = 4 6 P 4 = = = 360 ways  

27. EXAMPLE #3 Brian, Brenda, Brix, Brandon, and Brylle are to sit in the front seat of the auditorium, if there are only 2 seats left, how many ways can they choose and arrange 2 persons to sit in the front seat of the auditorium?

28. EXAMPLE #3 n = 5 r = 2 5 P 2 = = = 20 ways  

29. EXAMPLE #4 There are 7 members in a group. Three of them are to be appointed as president, vice president, and secretary. How many ways can one choose the president, the vice president, and the secretary from the group?

30. EXAMPLE #4 n = 7 r = 3 7 P 3 = = = 210 ways  

31. ANSWER THE FOLLOWING

32. PROBLEM #1 In how many ways can the letters of the word “COURAGE” be arranged?

33. PROBLEM #2 A family consist of 9 members, In how many ways can they arrange their seat in a round table?

34. PROBLEM #2 A family consist of 9 members, In how many ways can they arrange their seat in a round table?

35. PROBLEM #3 Suppose you have 7 pens, 3 of them are color blue, 2 are color red and 2 are color black, in how many ways can you arrange the pens in a pen case?

36. PROBLEM #4 How many arrangement can you make if there are 8 persons to be seated but there are only 5 seats available?

37. COMBINATION

38. COMBINATION A way of selecting r objects out of n objects where arrangement is not important. The set of the different combinations formed from n objects taken r at a time is a subset of the set of permutation.

39. COMBINATION FORMULA  

40. EXAMPLE #1 In how many ways can 5 fruits be selected from the 7 available fruits in making fruit salad? 7 C 5 = =   n = 7 r = 5

41. 7 C 5 = 7 C 5 = 21 ways  

42. EXAMPLE #2 In how many ways can 3 pizza toppings be selected from the 10 toppings suggested in the menu? 10 C 3 = =   n = 10 r = 3

43. 10 C 3 = 10 C 3 = 120 ways  

44. EXAMPLE #3 In how many ways can 7 students be selected among 12 students trying out for the school basketball varsity? 12 C 7 = =   n = 12 r = 7

45. 12 C 7 = 12 C 7 = 792 ways  

46. EXAMPLE #4 Suppose you are given an ordinary deck of playing cards. In how many ways can 5 cards be selected. 52 C 5 = =   n = 52 r = 5

47. 52 C 5 = 52 C 5 = 2, 598, 960  

48. SOLVING PERMUTATION & COMBINATION

49. PROBLEM #2 In how many ways can 9 students out of 14 students be chosen to be a part of the school glee club?

50. PROBLEM #3 In the class of 30 students, 4 of them are to be announced as the class valedictorian, salutatorian, first honor, and second honor. If all of them are qualified to the said titles, in how many ways can 4 students be selected?

51. COMBINATION

52. EXAMPLE #1 A math teacher is thinking of forming a committee that will oversee the celebration. He would like to select two boys and three girls from his class of 12 boys and 14 girls. How many combinations of two boys and 3 girls are possible?

53. EXAMPLE #2 Three colored paper strips are drawn at random from a container containing three red and five yellow strips. A. How many combinations of three colored paper strips can be drawn? B. How many combinations of three red paper strips can be drawn? C. How many combination of two yellow paper strips and a red paper strips can be drawn.