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The word ‘coincidence’ is defined as an event that might have been arranged though it was accidental in actuality. Most of us perceive life as a set of coincidences that lead us to pre-destined conclusions despite believing in a being who is free from the shackles of time and space. The question is that a being, for whom time and space would be nothing more than two more dimensions, wouldn’t it be rather disparaging to throw events out randomly and witness how the history unfolds (as a mere spectator)? Did He really arrange the events such that there is nothing accidental about their occurrence? Or are all the lives of all the living beings merely a result of a set of events that unfolded one after another without there being a chronological order?

To arrive at satisfactory answers to above questions we must steer this discourse towards the concept of conditional probability. That is the chance of something to happen given that an event has already happened. Though, the prior event need not to be related to the succeeding one but must be essential for it occurrence. Our minds as I believe are evolved enough to analyze a story and identify the point in time where the story has originated or the set of events that must have happened to ensure the specific conclusion of the story. To simplify the conundrum let us assume a hypothetical scenario where a man just became a pioneer in the field of actuarial science. Imagine him telling us his story in reverse. “I became what I am today by making all the right choices. The fact that I am what I am today is only a fact because I decided to leave firm X for firm Y. But I would have never been in Firm X if I had decided not to drop out of college. But I would have never been able to make such a difficult yet pivotal decision if I did not get advised by my ex-principle to do so. That important conversation wouldn’t…...

...permutation In mathematics, the notion of permutation is used with several slightly different meanings, all related to the act of permuting (rearranging) objects or values. Informally, a permutation of a set of objects is an arrangement of those objects into a particular order. For example, there are six permutations of the set {1,2,3}, namely (1,2,3), (1,3,2), (2,1,3), (2,3,1), (3,1,2), and (3,2,1). One might define an anagram of a word as a permutation of its letters. The study of permutations in this sense generally belongs to the field of combinatorics. The number of permutations of n distinct objects is n×(n − 1)×(n − 2)×...×2×1, which number is called "n factorial" and written "n!". Permutations occur, in more or less prominent ways, in almost every domain of mathematics. They often arise when different orderings on certain finite sets are considered, possibly only because one wants to ignore such orderings and needs to know how many configurations are thus identified. For similar reasons permutations arise in the study of sorting algorithms in computer science. In algebra and particularly in group theory, a permutation of a set S is defined as a bijection from S to itself (i.e., a map S → S for which every element of S occurs exactly once as image value). This is related to the rearrangement of S in which each element s takes the place of the corresponding f(s). The collection of such permutations form a symmetric group. The key to its structure is the......

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...Combinations and Permutations What's the Difference? In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: |[pic] |"My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be | | |"bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad. | | | | |[pic] |"The combination to the safe was 472". Now we do care about the order. "724" would not work, nor would "247". It has to be | | |exactly 4-7-2. | So, in Mathematics we use more precise language: |[pic] |If the order doesn't matter, it is a Combination. | |[pic] |If the order does matter it is a Permutation. | | |[pic] |So, we should really call this a "Permutation Lock"! | In other words: A Permutation is an ordered Combination. |[pic] |To help you to remember...

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...Combinations and Permutations What's the Difference? In English we use the word "combination" loosely, without thinking if the order of things is important. In other words: |[pic] |"My fruit salad is a combination of apples, grapes and bananas" We don't care what order the fruits are in, they could also be | | |"bananas, grapes and apples" or "grapes, apples and bananas", its the same fruit salad. | | | | |[pic] |"The combination to the safe was 472". Now we do care about the order. "724" would not work, nor would "247". It has to be | | |exactly 4-7-2. | So, in Mathematics we use more precise language: |[pic] |If the order doesn't matter, it is a Combination. | |[pic] |If the order does matter it is a Permutation. | | |[pic] |So, we should really call this a "Permutation Lock"! | In other words: A Permutation is an ordered Combination. |[pic] |To help you to remember...

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... Report page Share Share this 6.3 Probabilities Using Counting TechniquesThis is a featured page In a number of different situations, it is not easy to determine the outcomes of an event by counting them individually. Alternatively, counting techniques that involve permutations and combinations are helpful when calculating theoretical probabilities. This section will examine methods for determining theoretical probabilities of successive or multiple events. Permutation? or Combination? The following flow chart will help determine which formula is suitable for any given question. By simply following a series of "yes" or "no" questions, the appropriate formula can be determined. Flow Ex. 1 - Using Permutations: 6.3 Probabilities Using Counting Techniques - MDM4U1@FMG 6.3 Probabilities Using Counting Techniques - MDM4U1@FMG The specific outcome of Mike starting in lane 1 and the other two starting in lane 2 and lane 3 can only happen one way, so n(A) = 1. Therefore, 6.3 Probabilities Using Counting Techniques - MDM4U1@FMG The probability that Mike will start in the first lane next to his other brothers in lane 2 and 3 is approximately 0.00101. Ex. 1(a) - Using Permutations: Exactly Three People form a line at a grocery store. What is the probability that they will line up in descending order of age? (I.e. oldest, middle and youngest) →Solution using the blank like method: n(A): # of ways they will line up in descending order of age,......

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... T1 S2 T2 S3 Here n is 3, r is 2 For principle of counting we must have same number of options. In above if S3 and T2 is not allowed, principle of counting doesn’t work If n=5 and r=3 {A,B,C,D,E} How many different ways can we arrange of taking 3 letters at a time? 5 *4*3= 60 ways This is permutation of n different thing taken r at a time 60=(5*4*3*2*1)/(2*1) = 5!/2!=5!(5-3)!=n!/(n-r)! We are talking about linear arrangement not the circular one here nPr= filling r places by n different thing n=5 {A,B,C,D,E} r=3 {A,B,C}, {A,B,D}, {A,C,D}, {A,C,E}………….. [Note: Arrangement is related to permutation. If we are considered about place or position it is permutation question. Selecting is related to permutation. If we are not considered about place or position it is combination question.] nCr, C(n,r)= n!/[(n-r)!r!] n=5 {A,B,C,D,E} r=3 {A,B,C}, {A,B,D}, {A,B,E}, {A,C,D}, {A,C,E},{A,D,E}, {B,C,D},{B,C,E} 3 are selected out of 5 . Above are possible combinations. Permutation: {A,B,C} and {A,C,B} are different because here, in {A,B,C} second place we have B but in {A,C,B} second place we have C Combination: {A,B,C} and {A,C,B} are same because in both of them letters A,B,C are selected. {A,B,C} n=3 and r=3 then, with repetition no of arrangements = 3*3*3=27 ways but without repeatation number of ways = 3*2*1= 6 ways...

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...Page |1 PERMUTATIONS and COMBINATIONS If the order doesn't matter, it is a Combination. If the order does matter it is a Permutation. PRACTICE! Determine whether each of the following situations is a Combination or Permutation. 1. Creating an access code for a computer site using any 8 alphabet letters. 2. Determining how many different ways you can elect a Chairman and Co-Chairman of a committee if you have 10 people to choose from. 3. Voting to allow 10 new members to join a club when there are 25 that would like to join. 4. Finding different ways to arrange a line-up for batters on a baseball team. 5. Choosing 3 toppings for a pizza if there are 9 choices. Answers: 1. P 2. P 3. C 4. P 5. C Page |2 Combinations: Suppose that you can invite 3 friends to go with you to a concert. If you choose Jay, Ted, and Ken, then this is no different from choosing Ted, Ken, and Jay. The order that you choose the three names of your friends is not important. Hence, this is a Combination problem. Example Problem for Combination: Suppose that you can invite 3 friends to go with you to a concert. You have 5 friends that want to go, so you decide to write the 5 names on slips of paper and place them in a bowl. Then you randomly choose 3 names from the bowl. If the five people are Jay, Ted, Cal, Bob, and Ken, then write down all the possible ways that you could choose a group of 3 people. Here are all of the possible combinations of 3: Jay, Ted, Cal Jay, Ted, Bob Jay, Ted, Ken Jay,......

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...enumeration methods; sum rule, product rule, permutations, combinations along with enumeration methods for indistinguishable objects, how can we devise a strategy to solve problems requiring these methods? A basic concept in the branch of the theory of algorithms called enumeration theory, which investigates general properties of classes of objects numbered by arbitrary constructive objects (cf. Constructive object). Most often, natural numbers appear in the role of the constructive objects that serve as numbers of the elements of the classes in question ("Enumeration", 2013). The Sum/Difference Rules refer to the derivative of the sum of two functions is the sum of the derivatives of the two functions ("Basic Derivative Rules", 2013). The product rule is one of several rules used to find the derivative of a function. Specifically, it is used to find the derivative of the product of two functions. It is also called Leibnitz's Law, and it states that for two functions f and g their derivative (in Leibnitz notation, ). The derivative of f times g is not equal to the derivative of f times the derivative of g: .The product rule can be used with multiple functions and is used to derive the power rule. The product rule can also be applied to dot products and cross products of vector functions. The Leibnitz Identity, a generalization of the product rule, can be applied to find higher-order derivatives ("Definition Of Product Rule", 2013). A permutation in mathematics is one of......

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...Permutations Deﬁnition: A permutation of a set X is a rearrangement of its elements. Example: Let X = {1, 2, 3}. Then there are 6 permutations: 123, 132, 213, 231, 312, 321. Deﬁnition : A permutation of a set X is a one-one correspondence (a bijection) from X to itself. Notation: Let X = {1, 2, . . . , n} and α : X → X be a permutation. It is convenient to describe this function in the following way: α= 1 2 ... n α(1) α(2) . . . α(n) . Example: 1 2 2 1 1 2 3 1 2 3 1 2 3 2 3 1 1 2 3 4 1 4 3 2 1 2 3 4 5 3 5 4 1 2 Deﬁnition: Let X = {1, 2, . . . , n} and α : X → X be a permutation. Let i1 , i2 , . . . , ir be distinct numbers from {1, 2, . . . , n}. If α(i1 ) = i2 , α(i2 ) = i3 , . . . , α(ir−1 ) = ir , α(ir ) = i1 , and α(iν ) = iν for other numbers from {1, 2, . . . , n}, then α is called an r-cycle. Notation: An r-cycle is denoted by (i1 i2 . . . ir ). Example: 1 2 3 4 5 2 5 3 4 1 1 2 3 4 5 2 5 4 3 1 = (125) 3 − cycle is not a cycle 1 Remark: We can use diﬀerent notations for the same cycles. For example, 1 2 3 1 2 3 = (1) = (2) = (3), 1 2 3 2 3 1 = (123) = (231) = (312). Warning: Do not confuse notations of a permutation and a cycle. For example, (123) = 123. Instead, (123) = 231 and 123 = (1). Composition (Product) Of Permutations Let α= Then α◦β = β◦α= 1 2 ... n α(β(1)) α(β(2)) . . . α(β(n)) 1 2 ... n β(α(1)) β(α(2)) . . . β(α(n)) , . 1 2 ... n α(1) α(2) . . . α(n) and β = 1 2 ... n β(1) β(2) . . . β(n) . Example: Let α = 1 2 3 4 5 5 1 2 4 3......

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...PRG 410 WEEK 7 ASSIGNMENT http://www.coursehomework.com/product/prg-410-week-7-assignment/ Contact us at: +1 315-750-4434 help@coursehomework.com PRG 410 WEEK 7 ASSIGNMENT UOP C++ Programming I PRG-410 ASSIGNMENT # 5 Problem 1: Write a program that reads a base ten integer number as input and outputs the binary representation of that number. e.g. 29 must give as output Problem 2: Write a program that populates an array elements. Total number of permutations of a list of 'n' elements is n! (n factorial), so in this case total number of permutations will be 120. If input array = [17, 22, 14, 13], its all permutat Problem 3: Write a program to display all possible permutations of the digits of a given input integer. i.e. if an input integer is 432, then it has three digits 4, 3 and 2 and number of possible permutations is 6 (= 3! ). These permutations are 4 3 2 4 2 3 3 2 4 3 4 2 2 3 4 2 4 3 You may assume that the number < 10000. Click Here to Buy this; http://www.coursehomework.com/product/prg-410-week-7-assignment Course Home Work aims to provide quality study notes and tutorials to the students of PRG 410 Week 7 Assignment in order to ace their studies. Course Home Work - Best Home Work Tutorials PRG 410 Week 7 Assignment Course Home Work, PRG 410 Week 7 Assignment, Home Work Tutorials, Home Work Solutions, Home Work Essay, Home Work Questions.ACC 565 Wk 7 Assignment 3, ACC403 week 2 assignment, ACC565 Week 10, ACCT 212 (Financial...

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...Factorials, Permutations and Combinations Factorials A factorial is represented by the sign (!). When we encounter n! (known as 'n factorial') we say that a factorial is the product of all the whole numbers between 1 and n, where n must always be positive. For example 0! is a special case factorial. This is special because there are no positive numbers less than zero and we defined a factorial as a product of the numbers between n and 1. We say that 0! = 1 by claiming that the product of no numbers is 1. The reasoning and mathematics behind this is complicated and beyond the scope of this page, so let's just accept 0! as equal to 1. This works out to be mathematically true and allows us to redefine n! as follows: For example The above allows us to manipulate factorials and break them up, which is useful in combinations and permutations. Useful Factorial Properties The last two properties are important to remember. The factorial sign DOES NOT distribute across addition and subtraction. Permutations and Combinations Permutations and Combinations in mathematics both refer to different ways of arranging a given set of variables. Permutations are not strict when it comes to the order of things while Combinations are. For example; given the letters abc The Permutations are listed as follows Combinations on the other hand are considered different, all the above are considered the same since they have the exact same letters only arranged different. In other words, in......

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...number of miles the automobile can be driven without refueling. 13. To make a profit, a local store marks up the prices of its items by a certain percentage. Write a Java program that reads the original price of the item sold, the percentage of the marked-up price, and the sales tax rate. The program then outputs the original price of the item, the marked-up percentage of the item, the store’s selling price of the item, the sales tax rate, the sales tax, and the final price of the item. (The final price of the item is the selling price plus the sales tax.) 17. A permutation of three objects, a, b, and c, is any arrangement of these objects in a row. For example, some of the permutations of these objects are abc, bca, and cab. The number of permutations of three objects is 6. Suppose that these three objects are strings. Write a program that prompts the user to enter three strings. The program then outputs the six permutations of those strings. 19. Write a program that prompts the user to input the amount of rice, in pounds, in a bag. The program outputs the number of bags needed to store one metric ton of rice....

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...The principle of inclusion–exclusion will be used to count the permutations of n objects that leave no objects in their original positions. Consider Example 4. EXAMPLE 4 The Hatcheck Problem Anewemployee checks the hats of n people at a restaurant, forgetting to put claim check numbers on the hats. When customers return for their hats, the checker gives them back hats chosen at random from the remaining hats. What is the probability that no one receives the correct hat? ▲ Remark: The answer is the number of ways the hats can be arranged so that there is no hat in its original position divided by n!, the number of permutations of n hats.We will return to this example after we find the number of permutations of n objects that leave no objects in their original position. A derangement is a permutation of objects that leaves no object in its original position. To solve the problem posed in Example 4 we will need to determine the number of derangements of a set of n objects. EXAMPLE 5 The permutation 21453 is a derangement of 12345 because no number is left in its original position. However, 21543 is not a derangement of 12345, because this permutation leaves 4 fixed. ▲ Let Dn denote the number of derangements of n objects. For instance, D3 = 2, because the derangements of 123 are 231 and 312.We will evaluate Dn, for all positive integers n, using the principle of inclusion–exclusion. THEOREM 2 The number of derangements of a set with n elements is Dn = n! 1...

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...PRG 410 WEEK 7 ASSIGNMENT To purchase this Click here: http://www.activitymode.com/product/prg-410-week-7-assignment/ Contact us at: SUPPORT@ACTIVITYMODE.COM PRG 410 WEEK 7 ASSIGNMENT UOP C++ Programming I PRG-410 ASSIGNMENT # 5 Problem 1: Write a program that reads a base ten integer number as input and outputs the binary representation of that number. e.g. 29 must give as output Problem 2: Write a program that populates an array elements. Total number of permutations of a list of 'n' elements is n! (n factorial), so in this case total number of permutations will be 120. If input array = [17, 22, 14, 13], its all permutat Problem 3: Write a program to display all possible permutations of the digits of a given input integer. i.e. if an input integer is 432, then it has three digits 4, 3 and 2 and number of possible permutations is 6 (= 3! ). These permutations are 4 3 2 4 2 3 3 2 4 3 4 2 2 3 4 2 4 3 You may assume that the number < 10000. Activity Mode aims to provide quality study notes and tutorials to the students of PRG 410 Week 7 Assignment in order to ace their studies. PRG 410 Week 7 Assignment Activity Mode , PRG 410 Week 7 Assignment, Home Work Tutorials, Home Work Solutions, Home Work Essay, Home Work Questions.ACC 565 Wk 7 Assignment 3, ACC403 week 2 assignment, ACC565 Week 10, ACCT 212 (Financial Accounting), ACCT 344 (Entire Course) - Devry, ACCT 344 Final Exam Latest 2014 - Devry, ACCT 346 (Managerial Accounting), ACCT 346......

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...personality of the child. Learning activities are organized in terms of larger units. The teacher must also know her pupil’s like and dislikes. The child’s nature and experience must be made the starting points in planning and organizing school programs. Steps of the Integration Method. Subject: Probability Topic: Permutations and combinations 1. Introduction of the unit. Start off by explaining the objectives. After that, the teacher will present a pre-test about permutation that should be answered before the class ends. The teacher should correlate the lesson to the past lessons to show the relevance of the new one. 2. Point of experiencing. The teacher shows 5 different colored balls. The teacher will demonstrate how to compute permutations simply by counting the number of combinations she can get out of the 5 balls. After the demonstration: a. The teacher will show a formula that is easier to use to compute for permutations. b. The teacher will guide the students as to how to compute it using the manual method and with scientific calculator. c. The teacher will show a video clip about lotto draw, poker game and baseball game and solve permutation problems regarding the videos that was watched. 3. Culmination and evaluation of the unit. To further test whether the students fully grasp the objective of the unit being tackled, the teacher will conduct three activities. a. The teacher and the students will play a bingo game in which every draw, the students......

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...representation of that number. e.g. 29 must give as output Problem 2: Write a program that populates an array elements. Total number of permutations of a list of ‘n’ elements is n! (n factorial), so in this case total number of permutations will be 120. If input array = [17, 22, 14, 13], its all permutat Problem 3: Write a program to display all possible permutations of the digits of a given input integer. i.e. if an input integer is 432, then it has three digits 4, 3 and 2 and number of possible permutations is 6 (= 3! ). These permutations are 4 3 2 4 2 3 3 2 4 3 4 2 2 3 4 2 4 3 You may assume that the number < 10000. PRG 410 WEEK 7 ASSIGNMENT UOP C++ Programming I PRG-410 ASSIGNMENT # 5 Problem 1: Write a program that reads a base ten integer number as input and outputs the binary representation of that number. e.g. 29 must give as output Problem 2: Write a program that populates an array elements. Total number of permutations of a list of ‘n’ elements is n! (n factorial), so in this case total number of permutations will be 120. If input array = [17, 22, 14, 13], its all permutat Problem 3: Write a program to display all possible permutations of the digits of a given input integer. i.e. if an input integer is 432, then it has three digits 4, 3 and 2 and number of possible permutations is 6 (= 3! ). These permutations are 4 3 2 4 2 3 3 2 4 3 4 2 2 3 4 2 4 3 You may assume that the number < 10000. PRG 410 WEEK 7 ASSIGNMENT UOP C++ Programming......

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