Strings. A string is a list of symbols in a particular order.

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1 Ihor Stasyuk

2 Strings A string is a list of symbols in a particular order.

3 Strings A string is a list of symbols in a particular order. Examples: is a string of integers. X Q R A X P T is a string of letters from the usual alphabet. Any positive integer can be represented by a string of digits from the set {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}. Notice that the same symbol may appear more than once in a string.

4 Counting strings Example. Find the total number of possible words consisting of 5 letters.

5 Counting strings Example. Find the total number of possible words consisting of 5 letters. 1 st letter 2 nd letter 3 rd letter 4 th letter 5 th letter

6 Counting strings Example. Find the total number of possible words consisting of 5 letters. 26 ways 1 st letter 2 nd letter 3 rd letter 4 th letter 5 th letter

7 Counting strings Example. Find the total number of possible words consisting of 5 letters. 26 ways 26 ways 1 st letter 2 nd letter 3 rd letter 4 th letter 5 th letter

8 Counting strings Example. Find the total number of possible words consisting of 5 letters. 26 ways 26 ways 26 ways 1 st letter 2 nd letter 3 rd letter 4 th letter 5 th letter

9 Counting strings Example. Find the total number of possible words consisting of 5 letters. 26 ways 26 ways 26 ways 26 ways 1 st letter 2 nd letter 3 rd letter 4 th letter 5 th letter

10 Counting strings Example. Find the total number of possible words consisting of 5 letters. 26 ways 26 ways 26 ways 26 ways 26 ways 1 st letter 2 nd letter 3 rd letter 4 th letter 5 th letter

11 Counting strings Example. Find the total number of possible words consisting of 5 letters. 26 ways 26 ways 26 ways 26 ways 26 ways 1 st letter 2 nd letter 3 rd letter 4 th letter 5 th letter The total number of such words is = 26 5.

12 Counting strings Example. Find the number of 4-digit strings using digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} if no digit appears more than once in the string.

13 Counting strings Example. Find the number of 4-digit strings using digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} if no digit appears more than once in the string. 1 st digit 2 nd digit 3 rd digit 4 th digit

14 Counting strings Example. Find the number of 4-digit strings using digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} if no digit appears more than once in the string. 10 ways 1 st digit 2 nd digit 3 rd digit 4 th digit

15 Counting strings Example. Find the number of 4-digit strings using digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} if no digit appears more than once in the string. 10 ways 9 ways 1 st digit 2 nd digit 3 rd digit 4 th digit

16 Counting strings Example. Find the number of 4-digit strings using digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} if no digit appears more than once in the string. 10 ways 9 ways 8 ways 1 st digit 2 nd digit 3 rd digit 4 th digit

17 Counting strings Example. Find the number of 4-digit strings using digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} if no digit appears more than once in the string. 10 ways 9 ways 8 ways 7 ways 1 st digit 2 nd digit 3 rd digit 4 th digit

18 Counting strings Example. Find the number of 4-digit strings using digits {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} if no digit appears more than once in the string. 10 ways 9 ways 8 ways 7 ways 1 st digit 2 nd digit 3 rd digit 4 th digit The total number of such strings is = 5040.

19 Counting strings Example. Rolling a six sided die 3 times, how many possible outcomes are there with the first number 4 or 5 and the last number not 6?

20 Counting strings Example. Rolling a six sided die 3 times, how many possible outcomes are there with the first number 4 or 5 and the last number not 6? 1 st roll 2 nd roll 3 rd roll

21 Counting strings Example. Rolling a six sided die 3 times, how many possible outcomes are there with the first number 4 or 5 and the last number not 6? 2 results possible 1 st roll 2 nd roll 3 rd roll

22 Counting strings Example. Rolling a six sided die 3 times, how many possible outcomes are there with the first number 4 or 5 and the last number not 6? 2 results possible 6 results possible 1 st roll 2 nd roll 3 rd roll

23 Counting strings Example. Rolling a six sided die 3 times, how many possible outcomes are there with the first number 4 or 5 and the last number not 6? 2 results possible 6 results possible 5 results possible 1 st roll 2 nd roll 3 rd roll

24 Counting strings Example. Rolling a six sided die 3 times, how many possible outcomes are there with the first number 4 or 5 and the last number not 6? 2 results possible 6 results possible 5 results possible 1 st roll 2 nd roll 3 rd roll The total number of such outcomes is = 60.

25 Counting Strings The Multiplication Principle Suppose in a string of length k the first term can be chosen in n 1 ways, the second term can be chosen in n 2 ways regardless of how the first term has been filled,, the last k-th term can be chosen in n k ways regardless of how the previous terms have been filled. Then the total number of possible strings is n 1 n 2 n k. n 1 ways n 2 ways n k ways.. 1 st term 2 nd term k th term

26 Counting Strings Applying the Multiplication Principle, the total number of strings with k terms where each term can be chosen in n ways, is equal to n k. n ways n ways n ways st term 2 nd term k th term

27 Permutations A permutation of a collection of elements is an ordering of these elements in a row.

28 Permutations A permutation of a collection of elements is an ordering of these elements in a row. In other words, it is a string where each element of this collection appears exactly once.

29 Permutations A permutation of a collection of elements is an ordering of these elements in a row. In other words, it is a string where each element of this collection appears exactly once. Example. There are 6 possible permutations of the collection {a,b,c} : abc, acb, bac, cab, bca, cba.

30 Permutations Apply the Multiplication Principle to find the number of permutations of a collection with n elements : Collection of n elements. 1 st term 2 nd term n th term Ordering in a row

31 Permutations Apply the Multiplication Principle to find the number of permutations of a collection with n elements : Collection of n elements n ways. 1 st term 2 nd term n th term Ordering in a row

32 Permutations Apply the Multiplication Principle to find the number of permutations of a collection with n elements : Collection of n elements n ways n-1 ways. 1 st term 2 nd term n th term Ordering in a row

33 Permutations Apply the Multiplication Principle to find the number of permutations of a collection with n elements : Collection of n elements n ways n-1 ways 1 way. 1 st term 2 nd term n th term Ordering in a row

34 Permutations Apply the Multiplication Principle to find the number of permutations of a collection with n elements : Collection of n elements n ways n-1 ways 1 way Ordering in a row. 1 st term 2 nd term n th term The total number is n (n-1) 1 = n! n-factorial.

35 Permutations Example. In how many ways can 4 people be seated in a row of 4 theatre seats?

36 Permutations Example. In how many ways can 4 people be seated in a row of 4 theatre seats?

37 Permutations Example. In how many ways can 4 people be seated in a row of 4 theatre seats?

38 Permutations Example. In how many ways can 4 people be seated in a row of 4 theatre seats?

39 Permutations Example. In how many ways can 4 people be seated in a row of 4 theatre seats?

40 Permutations Example. In how many ways can 4 people be seated in a row of 4 theatre seats? The answer is 4! = = 24.

41 Combinations The term combination applies to a list of elements when the order of the elements is not taken into account.

42 Combinations The term combination applies to a list of elements when the order of the elements is not taken into account. Example. The two strings ABC and BCA both represent the same combination of the letters A, B and C.

43 Combinations The term combination applies to a list of elements when the order of the elements is not taken into account. Example. The two strings ABC and BCA both represent the same combination of the letters A, B and C. Example. ABCD is the same combination as ACDB but is different from ABCE.

44 Combinations Choosing k elements out of n Problem. In how many ways can we choose k elements out of n elements? (k n)

45 Combinations Choosing k elements out of n Problem. In how many ways can we choose k elements out of n elements? (k n) To get the answer, we have to find the number of combinations that consist of k distinct elements taken from a collection of n elements.

46 Combinations Choosing k elements out of n Collection of n elements n ways n-1 ways n-k+1 ways 1 st term 2 nd term k th term The number of ordered selections is: n ( n 1)... ( n k + 1) = n ( n 1)... ( n k + 1) ( n k) ( n k ( n k) ( n k 1) ) = (n n! k)! Since each combination can be rearranged in k! ways, we must divide by k! : n! k!( n k)!

47 Combinations So there are We denote n! k!( n k)! n! k!( n k)! ways of choosing k elements out of n elements. = n k n We read the symbol n choose k. k By definition, 0!=1, so, 0 = 0 0! 0!(0 0)! 1 = 1 1 = 1.

48 Combinations Example. Find the number of ways to choose 2 books out of 4.

49 Combinations Example. Find the number of ways to choose 2 books out of 4. k=2, n=4

50 Combinations Example. Find the number of ways to choose 2 books out of 4. k=2, n=4 We obtain : n! k!( n k)! = 4! 2!(4 2)! = = = 6.

51 Combinations Example. Find the number of ways to choose 2 books out of 4. k=2, n=4 We obtain : n! k!( n k)! = 4! 2!(4 2)! = = = 6.

52 Combinations Example. Find the number of ways to choose 2 books out of 4. k=2, n=4 We obtain : n! k!( n k)! = 4! 2!(4 2)! = = = 6.

53 Combinations Example. Find the number of ways to choose 2 books out of 4. k=2, n=4 We obtain : n! k!( n k)! = 4! 2!(4 2)! = = = 6.

54 Combinations Example. Find the number of ways to choose 2 books out of 4. k=2, n=4 We obtain : n! k!( n k)! = 4! 2!(4 2)! = = = 6.

55 Combinations Example. Find the number of ways to choose 2 books out of 4. k=2, n=4 We obtain : n! k!( n k)! = 4! 2!(4 2)! = = = 6.

56 Combinations Example. Find the number of ways to choose 2 books out of 4. k=2, n=4 We obtain : n! k!( n k)! = 4! 2!(4 2)! = = = 6.

57 Combinations Example. Find the number of ways to choose 2 books out of 4. k=2, n=4 We obtain : n! k!( n k)! = 4! 2!(4 2)! = = = 6. 6 combinations

58 Pascal s Triangle To build the triangle, start with 1 at the top, then continue placing numbers below it in a triangular pattern. Each number is just the two numbers above it added together (except for the edges, which are all 1).

59 Blaise Pascal ( ) mathematician, physicist, inventor, writer and philosopher

60 Pascal's triangle was known in China in the early 11th century through the work of the Chinese mathematician Jia Xian ( ). Yang Hui ( ) (Pascal s) triangle, as depicted by the Chinese using rod numerals.

61 Pascals Triangle Blaise Pascal s version of the triangle

62 Pascal s Triangle Horizontal Sums are powers of 2.

63 Pascal s Triangle

64 Pascal s Triangle Exponents of 11 :

65 Pascal s Triangle Pascal s Formula n + 1 k = n + k 1 n k for every positive integers n and k with k n.

66 Leonardo Bonacci, or Fibonacci, Leonardo of Pisa ( ) Liber Abaci Book of Calculations Mathematician

67 Fibonacci Numbers Problem. A pair of rabbits is put in a limited area. This pair of rabbits produces another pair each month. If the rabbits do not die, the question is : How many pairs of rabbits there would be? The answer from the book is this sequence of numbers : the Fibonacci sequence F n The sequence is defined recursively by the following rule: F 1 =1, F 2 =1 and F n =F n-1 +F n-2 for every integer n >2.

68 By adding diagonal numbers of the Pascal Triangle Fibonacci sequence can be obtained Pascal s Triangle Fibonacci sequence

69 Sets A set is a collection of elements or disjoint objects, considered as an object in its own right.

70 Sets A set is a collection of elements or disjoint objects, considered as an object in its own right. We use capital letters to denote sets and small letters to denote its elements.

71 Sets A set is a collection of elements or disjoint objects, considered as an object in its own right. We use capital letters to denote sets and small letters to denote its elements. aa AA means : the element a is contained in the set A.

72 Sets Examples. A={1,2,3,4,5} is the set of all integers from 1 to 5. B ={x real number 1 < x < 2} = (1,2) is the interval with end points 1 and 2. C={a, c, e} is the set that contains three elements: the letters a, c and e. A and C are finite sets while B is an infinite set.

73 Operations on Sets A B The union of two sets A and B is the set which is the set of all elements that are either in A or in B.

74 Operations on Sets The union of two sets A and B is the set which is the set of all elements that are either in A or in B. Example. If A={-3,5,7} and B={-4, 0, 5, 9} then A B = {-4, -3, 0, 5, 7, 9} A B

75 Operations on Sets The union of two sets A and B is the set which is the set of all elements that are either in A or in B. Example. If A={-3,5,7} and B={-4, 0, 5, 9} then A B = {-4, -3, 0, 5, 7, 9} If C={e, f, g, h} and D={i, j, k, l, m} then C D= {e, f, g, h, i, j, k, l, m}. A B

76 Operations on Sets A B The intersection of two sets A and B is the set which is the set of all elements that are both in A and in B.

77 Operations on Sets The intersection of two sets A and B is the set which is the set of all elements that are both in A and in B. Example. A B A B If A={-7,2,4,11} and B={-1,2,5,9,11} then ={2,11}.

78 Operations on Sets The intersection of two sets A and B is the set which is the set of all elements that are both in A and in B. Example. A B If A={-7,2,4,11} and B={-1,2,5,9,11} then ={2,11}. If C=(0,2) and D=(3,7) are intervals then C D = - the empty set that has no elements. A B

79 Operations on Sets The union of sets A 1, A 2,, A k is the set k i= 1 A i = A 1 A 2... either in A 1 or in A 2,, or in A k. A k of all elements that are

80 Operations on Sets The intersection of sets A 1, A 2,, A k is the set k i= 1 A i = A 1 A 2... of all elements that are simultaneously in A 1, in A 2,, and in A k. A k

81 Counting elements of disjoint sets For a set A of finitely many elements let N(A) be the number of elements in A. For instance, if A= { α, β, γ, δ, ε} then N(A)=5.

82 Counting elements of disjoint sets For a set A of finitely many elements let N(A) be the number of elements in A. For instance, if A= { α, β, γ, δ, ε} then N(A)=5. The Addition Principle Suppose a finite set A equals the union of k distinct pairwise disjoint subsets A 1, A 2,, A k. Then N(A)=N(A 1 )+N(A 2 )+ +N(A k )

83 Counting elements of disjoint sets Example. In some state all license plates consist of symbols from 4 to 6 chosen from the 26 letters of the alphabet together with the 10 digits 0-9 (together 36 symbols). A C D 9 Q 1 B T B How many license plates are possible if repetition of symbols is allowed?

84 Counting elements of disjoint sets The set of all plates can be partitioned into disjoint sets A, B, C containing of those of 4, 5 and 6 symbols: Plates with 4 symbols Plates with 5 symbols Plates with 6 symbols A B C The set of all plates is the union: A B C. The number of all plates is: N( A B C ) = N(A)+N(B)+N(C) (addition principle) = (multiplication principle)

85 Counting elements of disjoint sets When repetition of symbols is not allowed, the total number of such plates is: = = 2, 238, 928, 128

86 The Inclusion/Exclusion Rule A, B finite sets if. The Inclusion/Exclusion Rule for 2 or 3 sets. If A, B, C are any finite sets then and ) ( ) ( ) ( B N A N B A N + = A B = ) ( ) ( ) ( ) ( B A N B N A N B A N + = ) ( ) ( ) ( ) ( ) ( ) ( ) ( C B A N C B N C A N B A N C N B N A N = ) ( C B A N

87 The Inclusion/Exclusion Rule A B A B C C

88 The Inclusion/Exclusion Rule Example. A college conducted a survey to explore the academic interests of its students. They were asked to place checks beside the numbers of all statements that were true for them. Statement 1 I was on the honour roll last term. Statement 2 I belong to an academic club such as math club or Spanish club. Statement 3 I am majoring in at least two subjects. Out of a sample of 100 students, 28 checked #1, 26 checked #2 and 14 checked #3, 8 checked both #1 and #2, 4 checked both #1 and #3, 3 checked both #2 and #3, 2 checked all the three statements #1, #2 and #3.

89 The Inclusion/Exclusion Rule How many students checked at least one of the statements?

90 The Inclusion/Exclusion Rule How many students checked at least one of the statements? Answer: the number of students who checked at least one of the statements is ) ( ) ( ) ( ) ( ) ( ) ( ) ( = = D C H N D C N D H N C H N D N C N H N = ) ( D C H N

91 The Inclusion/Exclusion Rule How many students checked none?

92 The Inclusion/Exclusion Rule How many students checked none? Answer: the number of students who checked none is =45.

93 The Inclusion/Exclusion Rule Fill in the numbers for all eight regions of the diagram. Answer: ) ( ) ( ) ( ) ( = + = + D C H N D H N C H N H N ) ( ) ( ) ( ) ( = + = + D C H N D C N C H N C N ) ( ) ( ) ( ) ( = + = + D C H N D C N D H N D N ) ( ) ( = = D C H N C H N ) ( ) ( = = D C H N D H N ) ( ) ( = = D C H N D C N

94 THANK YOU

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