Permutations and combinations examples
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This document contains 13 examples of permutations and combinations problems from a Mathematics 4 class. It provides example problems involving passcode combinations, palindromes, paths on a grid, lottery number selection, triangles in an octagon, quiz answers, multiple choice exams, hamburger toppings, NBA championship series outcomes, poker hands, and donut flavors. The document is intended to illustrate different applications of permutations and combinations in mathematics.
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01_Probability of Simple Events.ppt
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This document provides a lesson on conditional probability that includes:
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3. A discussion of how conditional probability can inform decisions about driving while using a cell phone, health, and sports.
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There are several key concepts around permutations discussed in the document:
1) A permutation is an arrangement of objects in a particular order. The number of permutations of n objects taken r at a time is written as nPr.
2) Factorial notation is used to efficiently calculate permutations, where n! means the product of all integers down to 1.
3) The fundamental principle of counting states that if one event can occur m ways and another n ways, they can occur together in m*n ways. This extends to multiple events.
4) Permutations of alike objects divide the total permutations by the factorial of the number of alike objects.
5.5 permutations and combinations
The document defines permutations and combinations. A k-permutation is an ordered lineup of k objects taken from a set of n objects. The number of k-permutations of n objects is nPk = n!/(n-k)!. A k-combination is an unordered collection of k objects from a set of n objects. The number of k-combinations of n objects is nCk = n!/(n-k)!k!. Several examples are provided to illustrate calculating the number of permutations and combinations.
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Permutations involve arranging objects in a particular order, where order matters. There are a few different formulas to calculate the number of permutations depending on the situation. The basic formula for permutations of n objects taken r at a time is nPr = n!/(n-r)!. Some examples provided calculate the number of permutations when arranging letters, skiers in a competition, and people entering a cave.
The Fundamental Counting Principle
Subway and Starbucks advertise over 10,000 combinations of ways to customize sandwiches and drinks due to the many condiment and serving options. Companies can calculate combinations using (1) tree diagrams to show all outcomes or (2) the fundamental counting principle of multiplying the number of options at each choice point. Tree diagrams specifically show individual outcomes while the counting principle gives the total number of outcomes. Examples demonstrate using each method to find combinations in probability problems.
Linear equations in two variables
the rectangular coordinate system and midpoint formulas, linear equations in two variable, slope of a line, equation of a line, applications of linear equations and graphing
Sample space, events, outcomes, and experiments
Probability is a branch of mathematics that deals with describing how likely events are to occur. An experiment is a process with uncertain outcomes, while an outcome is any possible result. The sample space is the set of all possible outcomes. Fundamental counting principles state that if one event can occur in m ways, a second in n ways, and a third in p ways, the total number of outcomes is m x n x p. Probability is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
Probability Overview
There are two ways to count the number of possible outcomes of an experiment:
1) Using a tree diagram to list out all the combinations
2) Using the Fundamental Counting Principle, which involves multiplying the number of choices for each event together.
To calculate the probability of compound events (events made up of two or more simple events), you first determine if the events are independent or dependent. For independent events, the probability of one event does not affect the other event. You calculate the probability by multiplying the individual probabilities together.
11.5 Independent and Dependent Events
Identify whether events are independent or depdendent
Find the probability of independent and dependent events
Probability(mutually exclusive events)
The document contains solutions to 3 probability problems:
1) Finding the probability of rolling a die with a face that is a multiple of 3 or 5. The probability is 5/6.
2) Finding the probability that a randomly selected digit between 1-50 is a multiple of 10 or 11. The probability is 18/50.
3) Finding the probability that a randomly drawn card is a queen or king. The probability is 8/52.
Probability of Union of Two events
The document discusses mutually exclusive and non-mutually exclusive events. It provides examples to illustrate the difference, including examples involving drawing balls from a jar numbered 1-15 and rolling a die. It discusses how to calculate the probability of unions of events depending on whether they are mutually exclusive or not. Key points are that for mutually exclusive events, the probability of their union is the sum of their individual probabilities, while for non-mutually exclusive events it is the sum of their probabilities minus their intersection.
The Fundamental Counting Principle
This document explains the fundamental counting principle and permutations. It provides examples of how to use the principle to calculate the number of possible outcomes for events with multiple steps or choices. These include examples like counting meal combinations, sandwich varieties, possible license plates, phone numbers, test answers, and card selections. It discusses accounting for situations where items can or cannot be repeated between choices.
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This document discusses trigonometry and its key concepts. It defines trigonometry as a branch of mathematics concerning the study of triangles and the relationship between side lengths and angles. It notes that Hipparchus of Nicaea in the 2nd century BC is considered the father of trigonometry. The document outlines important trigonometric ratios like sine, cosine and tangent, and defines concepts like adjacent, opposite, and hypotenuse sides of a right triangle. It also discusses Pythagoras' theorem relating side lengths in a right triangle.
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The document defines important terms related to permutation and combination, including factorial, fundamental principle of counting, and different types of permutations and combinations. It provides examples of calculating permutations with and without repetition, as well as combinations with and without repetition. Formulas are given for each case. Permutations refer to arrangements where order matters, while combinations are arrangements where order does not matter.
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The document defines key concepts for classifying algebraic expressions, including:
- Monomials have one term, binomials have two terms, and trinomials have three terms.
- The degree of a polynomial with one variable is the highest exponent, and with multiple variables it is the highest sum of exponents.
- A polynomial can be classified by the number of terms and its degree. The leading coefficient is the coefficient of the highest degree term.
01_Probability of Simple Events.ppt
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Converse, contrapositive, inverse
The document discusses converse, inverse, and contrapositive statements of conditional (if-then) statements. It provides examples of converting statements to their converse, inverse, and contrapositive forms. It also discusses determining the truth value of predicates by substituting values for predicate variables.
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The document discusses different types of expressions and how to evaluate algebraic expressions. It defines an expression as not containing an equal sign, and describes two types of expressions - numerical expressions containing only numbers and symbols, and algebraic expressions containing numbers, symbols, and variables. The document provides examples of expressions and explains that a variable represents an unknown value. It also discusses how to evaluate expressions by substituting values for variables.
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This document contains a lesson plan for teaching factoring non-perfect trinomials in Math 8. The lesson plan outlines intended learning outcomes, learning content including subject matter and reference materials, learning experiences through various activities, an evaluation, and assignment. Students will learn to define trinomials, factor non-perfect square trinomials, and apply factoring trinomials to geometric figures through guided practice with algebra tiles and examples.
3. permutation and combination
The document discusses permutations and combinations. It defines permutations as arrangements where order is important, and combinations as groupings where order is not important. It provides examples of calculating permutations and combinations using factorial notation and formulas. The differences between permutations and combinations are outlined. Conditions that can be applied, such as objects being together or separated, are also discussed along with how to handle them in calculations.
4.4 probability of compound events
This document defines and provides examples of simple, compound, mutually exclusive and inclusive events. It explains that for mutually exclusive events, the probability of A or B equals the sum of the individual probabilities, while for inclusive events it equals the sum of the individual probabilities minus the probability they both occur. It then gives examples of calculating probabilities for various card draws and die rolls.
Conditional Probability
This document provides a lesson on conditional probability that includes:
1. Examples and formulas for calculating conditional probability
2. Practice problems solving for conditional probabilities in situations involving cards, dice, families, and committees
3. A discussion of how conditional probability can inform decisions about driving while using a cell phone, health, and sports.
12.4 probability of compound events
The document defines key probability terms like union, intersection, mutually exclusive events, and complement. It provides examples of calculating probabilities of compound events, such as the probability of event A or B. It explains that if events are mutually exclusive, the probability of their union is the sum of their individual probabilities. The document ends with practice problems calculating probabilities of compound events using these concepts.
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1) A permutation is an arrangement of objects in a particular order. The number of permutations of n objects taken r at a time is written as nPr.
2) Factorial notation is used to efficiently calculate permutations, where n! means the product of all integers down to 1.
3) The fundamental principle of counting states that if one event can occur m ways and another n ways, they can occur together in m*n ways. This extends to multiple events.
4) Permutations of alike objects divide the total permutations by the factorial of the number of alike objects.
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The document defines permutations and combinations. A k-permutation is an ordered lineup of k objects taken from a set of n objects. The number of k-permutations of n objects is nPk = n!/(n-k)!. A k-combination is an unordered collection of k objects from a set of n objects. The number of k-combinations of n objects is nCk = n!/(n-k)!k!. Several examples are provided to illustrate calculating the number of permutations and combinations.
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- Permutation refers to arrangements that consider order, while combination refers to selections where order does not matter.
- The number of permutations of n distinct objects taken r at a time is nPr = n!/(n-r)!, while the number of combinations is nCr = n!/r!(n-r)!.
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This powerpoint was used in my 7th and 8th grade classes to review the fundamental counting principle used in our probability unit. There are three independent practice problems at the end.
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statiscs and probability math college to help student
This document discusses statistics and probability concepts such as the fundamental counting principle, permutations, combinations, theoretical probability, and experimental probability. It provides examples of how to use these concepts to calculate the number of possible outcomes in probability experiments and real-world scenarios. For instance, it shows how to use permutations and combinations to determine the number of ways a student government can select officers from a group of people or how many combinations there are to draw a set of cubes from a bag. It also demonstrates calculating theoretical probabilities, such as the likelihood of rolling certain numbers on dice, and experimental probabilities based on data from trials.
(7) Lesson 9.5
To find the sample space and probability of having cereal A with strawberries:
1. Use the Fundamental Counting Principle to determine there are 8 possible outcomes when choosing a cereal and berry.
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This document provides examples and definitions for permutations and combinations. It begins by defining a permutation as an arrangement of objects in a specific order where order matters. A combination is a selection of objects without regard to order. The key rules are explained: for permutations, "n objects taken r at a time" is calculated as nPr = n!/(n-r)!, and for combinations "n objects taken r at a time" is calculated as nCr = n!/r!(n-r)!. Several examples are worked through for both permutations and combinations problems to illustrate how to set them up and calculate the number of possible outcomes. Special cases like repeating objects are also addressed.
Permutations and Combinations.pptx
This document provides an overview of permutations and combinations. It defines permutations as arrangements of objects in a specific order where order matters, and combinations as selections of objects without regard to order. The key formulas are presented: permutations use nPr and combinations use nCr. Several examples are worked out step-by-step for both permutations and combinations problems involving people, locations, and other real-world scenarios. Special cases like repeating elements are also addressed.
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The document provides 12 examples of counting problems using the fundamental counting principle. It explains that if one event can occur in n ways and a second in m ways, the total number of outcomes is n*m. The examples include counting combinations of digits, tournament placements, outfit combinations, license plates, dice rolls, classroom seating arrangements, true/false tests, multiple choice tests, and card shuffles.
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1. The document contains examples and formulas for permutations and combinations. It discusses linear permutations, distinguishable permutations, circular permutations, and ring permutations.
2. One example asks how many five-letter words can be formed from a set of vowels and consonants if they must alternate with no repetition.
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This mathematics lesson teaches students divisibility tests for 3 and 9. It begins with a review of previous divisibility rules for 2, 5, 4, and 10. Students then learn two new rules:
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This document contains instructions and questions about calculating permutations and combinations. It asks the reader to:
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statiscs and probability math college to help student
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Unit 6 presentation base ten equality form of a number with trainer notes 7.9.08
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Presentation 5 quantity magnitude and numeration january 2
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留信网认证的作用:
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5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
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→ 【关于价格问题（保证一手价格）
我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
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办理旧金山分校毕业证UCSF学位证毕业证offer【微信：95270640 】格式相对统一，各专业都有相应的模板。通常包括以下部分：
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The Unbelievable Tale of Dwayne Johnson Kidnapping: A Riveting Saga
Introduction
The notion of Dwayne Johnson kidnapping seems straight out of a Hollywood thriller. Dwayne "The Rock" Johnson, known for his larger-than-life persona, immense popularity. and action-packed filmography, is the last person anyone would envision being a victim of kidnapping. Yet, the bizarre and riveting tale of such an incident, filled with twists and turns. has captured the imagination of many. In this article, we delve into the intricate details of this astonishing event. exploring every aspect, from the dramatic rescue operation to the aftermath and the lessons learned.
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The Origins of the Dwayne Johnson Kidnapping Saga
Dwayne Johnson: A Brief Background
Before discussing the specifics of the kidnapping. it is crucial to understand who Dwayne Johnson is and why his kidnapping would be so significant. Born May 2, 1972, Dwayne Douglas Johnson is an American actor, producer, businessman. and former professional wrestler. Known by his ring name, "The Rock," he gained fame in the World Wrestling Federation (WWF, now WWE) before transitioning to a successful career in Hollywood.
Johnson's filmography includes blockbuster hits such as "The Fast and the Furious" series, "Jumanji," "Moana," and "San Andreas." His charismatic personality, impressive physique. and action-star status have made him a beloved figure worldwide. Thus, the news of his kidnapping would send shockwaves across the globe.
Setting the Scene: The Day of the Kidnapping
The incident of Dwayne Johnson's kidnapping began on an ordinary day. Johnson was filming his latest high-octane action film set to break box office records. The location was a remote yet scenic area. chosen for its rugged terrain and breathtaking vistas. perfect for the film's climactic scenes.
But, beneath the veneer of normalcy, a sinister plot was unfolding. Unbeknownst to Johnson and his team, a group of criminals had planned his abduction. hoping to leverage his celebrity status for a hefty ransom. The stage was set for an event that would soon dominate worldwide headlines and social media feeds.
The Abduction: Unfolding the Dwayne Johnson Kidnapping
The Moment of Capture
On the day of the kidnapping, everything seemed to be proceeding as usual on set. Johnson and his co-stars and crew were engrossed in shooting a particularly demanding scene. As the day wore on, the production team took a short break. providing the kidnappers with the perfect opportunity to strike.
The abduction was executed with military precision. A group of masked men, armed and organized, infiltrated the set. They created chaos, taking advantage of the confusion to isolate Johnson. Johnson was outnumbered and caught off guard despite his formidable strength and fighting skills. The kidnappers overpowered him, bundled him into a waiting vehicle. and sped away, leaving everyone on set in a state of shock and disbelief.
The Immediate Aftermath
The immediate aftermath of the Dwayne Johnson kidnappin
The Enigmatic Portrait, In the heart of a sleepy town
In the heart of a sleepy town nestled between rolling hills and whispering pines …
created with AI assistance…
原版制作(Mercer毕业证书)摩斯大学毕业证在读证明一模一样
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本公司拥有海外各大学样板无数，能完美还原。
1:1完美还原海外各大学毕业材料上的工艺：水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠。文字图案浮雕、激光镭射、紫外荧光、温感、复印防伪等防伪工艺。材料咨询办理、认证咨询办理请加学历顾问Q/微741003700
【主营项目】
一.毕业证【q微741003700】成绩单、使馆认证、教育部认证、雅思托福成绩单、学生卡等！
二.真实使馆公证(即留学回国人员证明,不成功不收费)
三.真实教育部学历学位认证（教育部存档！教育部留服网站永久可查）
四.办理各国各大学文凭(一对一专业服务,可全程监控跟踪进度)
如果您处于以下几种情况：
◇在校期间，因各种原因未能顺利毕业……拿不到官方毕业证【q/微741003700】
◇面对父母的压力，希望尽快拿到；
◇不清楚认证流程以及材料该如何准备；
◇回国时间很长，忘记办理；
◇回国马上就要找工作，办给用人单位看；
◇企事业单位必须要求办理的
◇需要报考公务员、购买免税车、落转户口
◇申请留学生创业基金
留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
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原版定制【微信:bwp0011】《(mu毕业证书)美国迈阿密大学牛津分校毕业证学历证书》【微信:bwp0011】成绩单 、雅思、外壳、留信学历认证永久存档查询，采用学校原版纸张、特殊工艺完全按照原版一比一制作（包括：隐形水印，阴影底纹，钢印LOGO烫金烫银，LOGO烫金烫银复合重叠，文字图案浮雕，激光镭射，紫外荧光，温感，复印防伪）行业标杆！精益求精，诚心合作，真诚制作！多年品质 ,按需精细制作，24小时接单,全套进口原装设备，十五年致力于帮助留学生解决难题，业务范围有加拿大、英国、澳洲、韩国、美国、新加坡，新西兰等学历材料，包您满意。
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留信网认证的作用:
1:该专业认证可证明留学生真实身份
2:同时对留学生所学专业登记给予评定
3:国家专业人才认证中心颁发入库证书
4:这个认证书并且可以归档倒地方
5:凡事获得留信网入网的信息将会逐步更新到个人身份内，将在公安局网内查询个人身份证信息后，同步读取人才网入库信息
6:个人职称评审加20分
7:个人信誉贷款加10分
8:在国家人才网主办的国家网络招聘大会中纳入资料，供国家高端企业选择人才
【关于价格问题（保证一手价格）】
我们所定的价格是非常合理的，而且我们现在做得单子大多数都是代理和回头客户介绍的所以一般现在有新的单子 我给客户的都是第一手的代理价格，因为我想坦诚对待大家 不想跟大家在价格方面浪费时间
对于老客户或者被老客户介绍过来的朋友，我们都会适当给一些优惠。
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AUT毕业证假文凭【微信95270640】购买（奥克兰理工大学毕业证成绩单硕士学历）Q微信95270640代办AUT学历认证留信网伪造奥克兰理工大学学位证书精仿奥克兰理工大学本科/硕士文凭证书补办奥克兰理工大学 diplomaoffer,Transcript购买奥克兰理工大学毕业证成绩单购买AUT假毕业证学位证书购买伪造奥克兰理工大学文凭证书学位证书,专业办理雅思、托福成绩单，学生ID卡，在读证明，海外各大学offer录取通知书，毕业证书，成绩单，文凭等材料:1:1完美还原毕业证、offer录取通知书、学生卡等各种在读或毕业材料的防伪工艺（包括 烫金、烫银、钢印、底纹、凹凸版、水印、防伪光标、热敏防伪、文字图案浮雕，激光镭射，紫外荧光，温感光标）学校原版上有的工艺我们一样不会少，不论是老版本还是最新版本，都能保证最高程度还原，力争完美以求让所有同学都能享受到完美的品质服务。
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Permutations and combinations examples
- 1. Examples on Permutations and Combinations Mathematics 4 February 23, 2012 Mathematics 4 Examples on Permutations and Combinations
- 2. Example A stalker classmate observed that her crush’s smartphone has fingerprint smudges corresponding to the numbers 2,4,7, and 9 on the passcode entry screen. She correctly figures out that her crush’s passcode must be using those four digits. What is the largest possible number of trials that she needs to take to access the contents of the smartphone? Mathematics 4 Examples on Permutations and Combinations
- 3. Example How many 7 digit numerical palindromes are there? (Example: 34543) Mathematics 4 Examples on Permutations and Combinations
- 4. Example Given the grid below, how many paths from A to B are possible if moves are restricted to the right or up? Mathematics 4 Examples on Permutations and Combinations
- 5. Example How many ways can a player select six numbers in the Lotto 6/42 game? Mathematics 4 Examples on Permutations and Combinations
- 6. Example How many different triangles can be formed by connecting any 3 points in the vertices of a regular octagon? Mathematics 4 Examples on Permutations and Combinations
- 7. Example How many different triangles can be formed by connecting any 3 points in the vertices of a regular octagon? Mathematics 4 Examples on Permutations and Combinations
- 8. Example How many ways can a student answer 10 questions correctly in a 15 item quiz? Mathematics 4 Examples on Permutations and Combinations
- 9. Example If each test item in the quiz above is a multiple choice question with 4 choices each, how many ways can a student answer the exam? Mathematics 4 Examples on Permutations and Combinations
- 10. Example If each test item in the quiz above is a multiple choice question with 4 choices each, how many ways can a student make 5 mistakes in the exam? Mathematics 4 Examples on Permutations and Combinations
- 11. Example A customer in a fast food restaurant can choose any combination out of 7 condiments on a hamburger. How many burger sandwich combinations can the customer make? Mathematics 4 Examples on Permutations and Combinations
- 12. Example The winner of the seven-game NBA championship series is the team that wins four games. In how many different ways can the series finish in seven games? Mathematics 4 Examples on Permutations and Combinations
- 13. Example A full house is a hand that consists of three cards of one denomination and two cards of another denomination. How many 5-card hands are full houses? Mathematics 4 Examples on Permutations and Combinations
- 14. Example How many ways can we buy 12 donuts out of an unlimited supply of 5 flavors? Mathematics 4 Examples on Permutations and Combinations