1. Chapter 2 Combinatorial Analysis 主講人 : 虞台文

2. Content Unordered Samples without Replacement  Combinations Binomial Coefficients Some Useful Mathematic Expansions Unordered Samples with Replacement Derangement Calculus

3. Chapter 2 Combinatorial Analysis Unordered Samples without Replacement  Combinations

4. Combinations n distinct objects Choose k objects How many choices?

5. Combinations Drawing k objects, their order is unnoted, among n distinct objects w/o replacement, the number of possible outcomes is This notation is preferred

6. More on

7. Examples

8. Example 4 The mathematics department consists of 25 full professors, and 15 associate professors, and 35 assistant professors. A committee of 6 is selected at random from the faculty of the department. Find the probability that all the members of the committee are assistant professors. x Denoting the all-assistant event as E,

9. Example 5 A poker hand has five cards drawn from an ordinary deck of 52 cards. Find the probability that the poker hand has exactly 2 kings. x Denoting the 2-king event as E,

10. Example 6 Two boxes both have r balls numbered 1, 2, …, r. Two random samples of size m and n are drawn without replacement from the 1 st and 2 nd boxes, respectively. Find the probability that these two samples have exactly k balls with common numbers. 1 1 2 2 3 3 r r 1 1 2 2 3 3 r r m n P(“k matches”) = ? E |  |=? |E|=?

11. Example 6 1 1 2 2 3 3 r r 1 1 2 2 3 3 r r m n # possible outcomes from the 1 st box. # possible k -matches. # possible outcomes from the 2nd box for each k -match.

12. Example 6 1 1 2 2 3 3 r r 1 1 2 2 3 3 r r m n

13. 1 1 2 2 3 3 r r 1 1 2 2 3 3 r r m n 樂透和本例有何關係 ?

14. Example 6 1 1 2 2 3 3 r r 1 1 2 2 3 3 r r m n 本式觀念上係由第一口箱子出發所推得

15. Example 6 1 1 2 2 3 3 r r 1 1 2 2 3 3 r r m n 觀念上，若改由第二口箱子出發結果將如何 ?

16. Example 6 1 1 2 2 3 3 r r 1 1 2 2 3 3 r r m n

17. Exercise 1 1 2 2 3 3 r r 1 1 2 2 3 3 r r m n

18. Chapter 2 Combinatorial Analysis Binomial Coefficients

21. n terms

22. Binomial Coefficients n boxes

23. Binomial Coefficients Facts:

24. Properties of Binomial Coefficients

27. Exercise

28. Properties of Binomial Coefficients 第一類取法 :   第二類取法 :

29. Properties of Binomial Coefficients Pascal Triangular

30. Properties of Binomial Coefficients Pascal Triangular 1 1 1 1 1 1 1 1 1 2 3 3 4 6 4

31. Properties of Binomial Coefficients 吸星大法

32. Example 7-1

33. Example 7-2 k  x+1 Fact: ?

34. Example 7-2 k  k+1 簡化版

35. Example 7-3 k  k+2 簡化版 ?

36. Negative Binomial Coefficients

37. How to memorize? k k k (n)(n) 11

38. Negative Binomial Coefficients 這公式真的對嗎 ? k k k (n+k1)(n+k1) 11 1

39. Negative Binomial Coefficients

40. Chapter 2 Combinatorial Analysis Some Useful Mathematic Expansions

48. z 值沒有任何限制

49. Some Useful Mathematic Expansions

50. Chapter 2 Combinatorial Analysis Unordered Samples with Replacement

51. Discussion 投返 非投返 有序 無序 ?

52. Unordered Samples with Replacement n 不同物件任取 k 個 可重複選取 n 不同物件，每一 中物件均無窮多個 從其中任取 k 個

53. Unordered Samples with Replacement 此多項式乘開後 z k 之係數有何意義 ?

54. Unordered Samples with Replacement

55. 投返 非投返 有序 無序

56. Example 8 Suppose there are 3 boxes which can supply infinite red balls, green balls, and blue balls, respectively. How many possible outcomes if ten balls are chosen from them? n = 3 k = 10

57. Example 9 There are 3 boxes, the 1st box contains 5 red balls, the 2nd box contains 3 green balls, and the 3rd box contains infinite many blue balls. How many possible outcomes if k balls are chosen from them. k=1 有幾種取法 k=2 有幾種取法 k=3 有幾種取法 k=4 有幾種取法 觀察 :

58. Example 9 此多項式乘開後 z k 之係數卽為解

59. Example 9 此多項式乘開後 z k 之係數卽為解

60. Example 9 此多項式乘開後 z k 之係數卽為解 Coef(z k )=?

61. Example 9 Coef(z k )=? jk4jk4 jk6jk6 j  k  10 jkjk

62. Example 9 Coef(z k )=? jk4jk4 jk6jk6 j  k  10 jkjk

63. Example 9

65. Chapter 2 Combinatorial Analysis Derangement

66. 最後 ! ! ! 每一個人都拿 到別人的帽子 錯排

67. Example 10 n 人中正好 k 人拿 對自己的帽子 n 人中無人拿 對自己的帽子

68. Example 10

69.  n 人中正好 k 人拿對自己的帽子  n 人中無人拿對自己的帽子 12 21 12 23 3 1312 1/2!2/3!

70. Example 10  n 人中正好 k 人拿對自己的帽子  n 人中無人拿對自己的帽子 令 A i 表第 i 個人拿了自己帽子

71. Example 10 A i 表第 i 個人拿了自己帽子

72. Example 10 A i 表第 i 個人拿了自己帽子 12n 1...

73. Example 10 A i 表第 i 個人拿了自己帽子 12n 12...

74. Example 10 A i 表第 i 個人拿了自己帽子

75. Example 10 A i 表第 i 個人拿了自己帽子

76. Example 10 A i 表第 i 個人拿了自己帽子

77. Example 10 A i 表第 i 個人拿了自己帽子

78. Example 10 A i 表第個人拿了自己帽子

79. Example 10

80. ... k matches n  k mis matches 

81. Example 10

82. Remark

83. Chapter 2 Combinatorial Analysis Calculus

84. Some Important Derivatives Derivatives for multiplications — Derivatives for divisions — Chain rule —

85. L’Hopital rule

86. Examples

87. Integration by Part

89. The Gamma Function

90. Example 12

93. 0  (   1)

94. Example 12

99. 