12.5 permutations 1
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The document discusses permutations and combinations. It provides examples of using permutations to calculate the number of possible arrangements or orderings of objects. This includes examples such as arranging letters, choosing batting orders, and selecting student government positions. Formulas are provided for calculating permutations, including using factorials. Practice problems are provided to help understand evaluating permutations.
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12.5 permutations 1
This document contains lesson materials on permutations and combinations including examples and explanations. It discusses key concepts such as:
- Permutations involve arrangements where order matters
- The factorial operation (!) is used to write the product of consecutive whole numbers
- The permutation formula nPr is used to calculate the number of arrangements of r objects from a set of n objects
- Examples are provided to demonstrate calculating permutations and combinations in different scenarios like arranging letters, choosing swimmers for lanes, and selecting student government positions.
Probability
The document provides an overview of lessons on probability for Mathematics 10 covering weeks 6-8 of the third quarter. Key concepts covered include terms related to probability, probability of simple and compound events, probability of the union and intersection of events, conditional probability, and examples of sample spaces and events. Some common phrases involving numbers are also defined.
CPSC 125 Ch 3 Sec 4 6
The document discusses permutations and combinations. Some key points:
1. A permutation is an ordered arrangement of objects. The number of permutations of n objects taken r at a time is written as P(n,r) or nPr.
2. A combination is an unordered selection of objects. The number of combinations of n objects taken r at a time is written as C(n,r) or nCr.
3. Examples are provided to illustrate calculating permutations and combinations in different scenarios like selecting committees, poker hands, and routes on a grid.
Permutation
This document provides an overview of permutations including definitions, examples of calculating permutations using factorials and the nPr notation, and applications of permutations such as arranging objects, selecting a subset of objects, and circular arrangements. It discusses permutations of n objects, distinguishable permutations, and permutations of n objects taken r at a time. Examples are provided to illustrate calculating permutations in different scenarios as well as applications and non-applications of permutations.
Permutation
This document discusses fundamental principles of counting and permutation and combination. It provides examples to illustrate key concepts such as:
1) Mohan has 3 pants and 2 shirts - there are 3×2 = 6 ways to choose a pant and shirt combination.
2) Sabnam has 2 bags, 3 boxes, and 2 bottles - the number of ways to choose one of each is the number of ways to successively choose a bag, box, and bottle.
3) With 4 roads between a student's house and college, there are 4×3 = 12 ways for the student to go to college and return on different roads.
Multiplication
Multiplication can be represented as repeated addition. The document provides examples of using multiplication to solve one-step word problems involving finding the total number of objects from a given number of groups containing the same number of objects in each group. Examples show setting up multiplication equations to represent 4 groups of 3 balls, 6 groups of 3, 4 groups of 2, and 2 groups of 5.
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This document provides examples of counting techniques problems involving permutations, combinations, and arrangements. It includes 17 questions on line permutations, circular permutations, pair permutations, and combinations. The questions cover a range of scenarios involving selecting committees, arranging objects, seating arrangements, and more.
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This document provides an explanation of permutations and combinations in 5 parts. It introduces the fundamental counting principle and factorial notation. It discusses arrangements with and without repetition, as well as specific positions. Examples are provided for each concept, such as the number of arrangements of letters in a word or songs on a playlist. Questions assess understanding with additional word problems and step-by-step solutions are provided.
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12.5 permutations 1
This document contains lesson materials on permutations and combinations including examples and explanations. It discusses key concepts such as:
- Permutations involve arrangements where order matters
- The factorial operation (!) is used to write the product of consecutive whole numbers
- The permutation formula nPr is used to calculate the number of arrangements of r objects from a set of n objects
- Examples are provided to demonstrate calculating permutations and combinations in different scenarios like arranging letters, choosing swimmers for lanes, and selecting student government positions.
Probability
The document provides an overview of lessons on probability for Mathematics 10 covering weeks 6-8 of the third quarter. Key concepts covered include terms related to probability, probability of simple and compound events, probability of the union and intersection of events, conditional probability, and examples of sample spaces and events. Some common phrases involving numbers are also defined.
CPSC 125 Ch 3 Sec 4 6
The document discusses permutations and combinations. Some key points:
1. A permutation is an ordered arrangement of objects. The number of permutations of n objects taken r at a time is written as P(n,r) or nPr.
2. A combination is an unordered selection of objects. The number of combinations of n objects taken r at a time is written as C(n,r) or nCr.
3. Examples are provided to illustrate calculating permutations and combinations in different scenarios like selecting committees, poker hands, and routes on a grid.
Permutation
This document provides an overview of permutations including definitions, examples of calculating permutations using factorials and the nPr notation, and applications of permutations such as arranging objects, selecting a subset of objects, and circular arrangements. It discusses permutations of n objects, distinguishable permutations, and permutations of n objects taken r at a time. Examples are provided to illustrate calculating permutations in different scenarios as well as applications and non-applications of permutations.
Permutation
This document discusses fundamental principles of counting and permutation and combination. It provides examples to illustrate key concepts such as:
1) Mohan has 3 pants and 2 shirts - there are 3×2 = 6 ways to choose a pant and shirt combination.
2) Sabnam has 2 bags, 3 boxes, and 2 bottles - the number of ways to choose one of each is the number of ways to successively choose a bag, box, and bottle.
3) With 4 roads between a student's house and college, there are 4×3 = 12 ways for the student to go to college and return on different roads.
Multiplication
Multiplication can be represented as repeated addition. The document provides examples of using multiplication to solve one-step word problems involving finding the total number of objects from a given number of groups containing the same number of objects in each group. Examples show setting up multiplication equations to represent 4 groups of 3 balls, 6 groups of 3, 4 groups of 2, and 2 groups of 5.
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This document provides examples of counting techniques problems involving permutations, combinations, and arrangements. It includes 17 questions on line permutations, circular permutations, pair permutations, and combinations. The questions cover a range of scenarios involving selecting committees, arranging objects, seating arrangements, and more.
Pre calculus math 40s - permutations & combinations - lesson 1
This document provides an explanation of permutations and combinations in 5 parts. It introduces the fundamental counting principle and factorial notation. It discusses arrangements with and without repetition, as well as specific positions. Examples are provided for each concept, such as the number of arrangements of letters in a word or songs on a playlist. Questions assess understanding with additional word problems and step-by-step solutions are provided.
12.5 permutations 2
The document provides instruction and examples for using permutations and combinations to count possible outcomes. It defines permutations and explains how to calculate them using factorials and the counting principle. Examples include counting the number of ways piano players and singers can be chosen, movies and snacks paired, and runners finishing in different places. Practice problems have students count orders of movies watched and ways bands can be ranked. Guidance is given for online calculators to solve permutation and combination problems.
6. Permutations and Combinations-Revised (1).pptx
There are three key points summarized:
1) A permutation is an arrangement of things in a certain order where order matters and repetition is not allowed. The number of permutations of n things taken r at a time can be calculated as nPr = n!/(n-r)!.
2) Examples are provided to demonstrate calculating permutations in different scenarios such as arranging books on a shelf or selecting people for roles.
3) Circular permutations are also discussed, where the number of ways to seat 6 people around a circular table with no distinction between rotations is 120.
12.6 combinations 1
This document provides instruction on combinations and permutations. It begins with examples of evaluating combinations and permutations, such as finding the number of combinations when choosing 3 items from a group of 8 items (8C3) or the number of permutations when selecting a preferred and substitute color from 12 color options (12P2). It then provides practice problems for students to determine whether a scenario involves combinations or permutations and write the appropriate expression to calculate the number of possibilities. The document aims to teach students the differences between combinations and permutations and how to set up and evaluate expressions to solve combinatorial problems.
Combinations and permutations(1)
The document discusses permutations and combinations in the context of seating arrangements. It provides examples to determine the number of possible arrangements for seating groups of people. For a group of 3 people, there are 6 possible arrangements. For a group of 5 people, there are 120 possible arrangements, found by multiplying 5 x 4 x 3 x 2 x 1. The document also discusses using permutations and combinations to solve other types of problems involving arrangements and selections where order does or does not matter.
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.
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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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.
counting techniques
The document discusses permutations and combinations. It provides examples of calculating permutations and combinations for different scenarios like selecting committees from a group of people and arranging books on a shelf. Formulas for permutations (nPr) and combinations (nCr) are given. Order matters for permutations but not for combinations. The key difference between the two is explained.
Probability quiz 1
The document provides two word problems involving combinations and permutations. The first problem asks how many apartment choices Cedric has if he can choose from three layouts each with or without a washer/dryer. The second problem asks how many channel and subtitle choices there are on a DVD that has three audio channels and four language subtitles. The document also reviews key concepts around permutations, combinations, probability and provides examples to practice these concepts.
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This document defines permutations and combinations and provides examples of how to calculate them. It begins by defining a permutation as an arrangement of objects in a specific order, with order mattering. A combination is a selection of objects without regard to order. The document then provides the formulas for calculating permutations and combinations using factorials. It gives examples such as finding the number of ways to arrange people in a picture or rank locations. It also explains how to calculate permutations and combinations on a graphing calculator.
Permutations and combinations
This material is a part of PGPSE / CSE study material for the students of PGPSE / CSE students. PGPSE is a free online programme for all those who want to be social entrepreneurs / entrepreneurs
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This material is for PGPSE / CSE students of AFTERSCHOOOL. PGPSE / CSE are free online programme - open for all - free for all - to promote entrepreneurship and social entrepreneurship PGPSE is for those who want to transform the world. It is different from MBA, BBA, CFA, CA,CS,ICWA and other traditional programmes. It is based on self certification and based on self learning and guidance by mentors. It is for those who want to be entrepreneurs and social changers. Let us work together. Our basic idea is that KNOWLEDGE IS FREE & AND SHARE IT WITH THE WORLD
Permutations and combinations
The document discusses various concepts related to permutations and combinations including:
1) It provides examples to explain the difference between permutations and combinations.
2) It lists the formula for permutations as Npr = n! / (n-r)! and provides an example calculation.
3) It poses several word problems involving permutations and combinations and shows the calculations to arrive at the answers.
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This document introduces concepts of counting theory including permutations and combinations. It provides examples of using tree diagrams, organized lists, and formulas to count outcomes of trials where order does or does not matter. Key examples include tossing coins, arranging books on a shelf, and forming committees. The objectives are to use diagrams and lists to enumerate outcomes, identify when to use permutations or combinations, and use formulas and a calculator to solve counting problems. Homework is assigned to create two word problems applying permutations and combinations.
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This document provides examples and explanations of tree diagrams, the counting principle, and probability. It includes examples of using tree diagrams to represent choices and outcomes, such as combinations of bread, meat and cheese for subs. It also demonstrates using the counting principle to calculate the total number of possible outcomes when multiple choices are involved, such as the number of possible outfits from different clothing items or passwords with letter and number options. The document concludes with an example of calculating probability as the ratio of favorable to total possible outcomes, such as the probability of four people rolling the same number on dice.
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This document discusses probability and counting principles such as permutations, combinations, and the fundamental counting principle. It provides examples of how to use formulas and tree diagrams to calculate the number of possible outcomes for compound events. Key topics covered include distinguishing between combinations and permutations, using factorials to find the number of arrangements and groupings, and solving probability problems through listing outcomes and applying counting rules.
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This document contains a mathematics lesson on combinations and permutations. It begins with examples of finding the number of combinations and permutations of letters. It then provides examples of combination word problems involving selecting groups from larger sets, forming committees, handshakes, and selecting exam questions. Applications of combinations in real world scenarios are also discussed, such as lotteries, fruit salads, and polygon formation using points.
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This document discusses permutations and combinations. It provides formulas for calculating the number of permutations and combinations of n objects taken r at a time. Examples are given to demonstrate calculating permutations and combinations to solve problems involving arranging objects in different orders or selecting objects without regard to order. Practice problems are included for students to calculate unknown values and solve word problems involving permutations and combinations.
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This document contains 15 multiple choice questions testing concepts in permutations and combinations. Permutations refer to arrangements of distinguishable objects without repetition, while combinations refer to subsets of objects without regard to order. The document also provides examples and formulas for calculating permutations and combinations, including cases where some objects are identical. It defines key permutation terms like factorial and P(n,r) and illustrates their use through examples like finding the number of ways to arrange 5 people for a portrait.
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12.5 permutations 2
The document provides instruction and examples for using permutations and combinations to count possible outcomes. It defines permutations and explains how to calculate them using factorials and the counting principle. Examples include counting the number of ways piano players and singers can be chosen, movies and snacks paired, and runners finishing in different places. Practice problems have students count orders of movies watched and ways bands can be ranked. Guidance is given for online calculators to solve permutation and combination problems.
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There are three key points summarized:
1) A permutation is an arrangement of things in a certain order where order matters and repetition is not allowed. The number of permutations of n things taken r at a time can be calculated as nPr = n!/(n-r)!.
2) Examples are provided to demonstrate calculating permutations in different scenarios such as arranging books on a shelf or selecting people for roles.
3) Circular permutations are also discussed, where the number of ways to seat 6 people around a circular table with no distinction between rotations is 120.
12.6 combinations 1
This document provides instruction on combinations and permutations. It begins with examples of evaluating combinations and permutations, such as finding the number of combinations when choosing 3 items from a group of 8 items (8C3) or the number of permutations when selecting a preferred and substitute color from 12 color options (12P2). It then provides practice problems for students to determine whether a scenario involves combinations or permutations and write the appropriate expression to calculate the number of possibilities. The document aims to teach students the differences between combinations and permutations and how to set up and evaluate expressions to solve combinatorial problems.
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The document discusses permutations and combinations in the context of seating arrangements. It provides examples to determine the number of possible arrangements for seating groups of people. For a group of 3 people, there are 6 possible arrangements. For a group of 5 people, there are 120 possible arrangements, found by multiplying 5 x 4 x 3 x 2 x 1. The document also discusses using permutations and combinations to solve other types of problems involving arrangements and selections where order does or does not matter.
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.
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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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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.
counting techniques
The document discusses permutations and combinations. It provides examples of calculating permutations and combinations for different scenarios like selecting committees from a group of people and arranging books on a shelf. Formulas for permutations (nPr) and combinations (nCr) are given. Order matters for permutations but not for combinations. The key difference between the two is explained.
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The document provides two word problems involving combinations and permutations. The first problem asks how many apartment choices Cedric has if he can choose from three layouts each with or without a washer/dryer. The second problem asks how many channel and subtitle choices there are on a DVD that has three audio channels and four language subtitles. The document also reviews key concepts around permutations, combinations, probability and provides examples to practice these concepts.
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This document defines permutations and combinations and provides examples of how to calculate them. It begins by defining a permutation as an arrangement of objects in a specific order, with order mattering. A combination is a selection of objects without regard to order. The document then provides the formulas for calculating permutations and combinations using factorials. It gives examples such as finding the number of ways to arrange people in a picture or rank locations. It also explains how to calculate permutations and combinations on a graphing calculator.
Permutations and combinations
This material is a part of PGPSE / CSE study material for the students of PGPSE / CSE students. PGPSE is a free online programme for all those who want to be social entrepreneurs / entrepreneurs
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The document discusses various concepts related to permutations and combinations including:
1) It provides examples to explain the difference between permutations and combinations.
2) It lists the formula for permutations as Npr = n! / (n-r)! and provides an example calculation.
3) It poses several word problems involving permutations and combinations and shows the calculations to arrive at the answers.
CP Power Point.ppt
This document introduces concepts of counting theory including permutations and combinations. It provides examples of using tree diagrams, organized lists, and formulas to count outcomes of trials where order does or does not matter. Key examples include tossing coins, arranging books on a shelf, and forming committees. The objectives are to use diagrams and lists to enumerate outcomes, identify when to use permutations or combinations, and use formulas and a calculator to solve counting problems. Homework is assigned to create two word problems applying permutations and combinations.
12.4 counting methods 1
This document provides examples and explanations of tree diagrams, the counting principle, and probability. It includes examples of using tree diagrams to represent choices and outcomes, such as combinations of bread, meat and cheese for subs. It also demonstrates using the counting principle to calculate the total number of possible outcomes when multiple choices are involved, such as the number of possible outfits from different clothing items or passwords with letter and number options. The document concludes with an example of calculating probability as the ratio of favorable to total possible outcomes, such as the probability of four people rolling the same number on dice.
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This document discusses probability and counting principles such as permutations, combinations, and the fundamental counting principle. It provides examples of how to use formulas and tree diagrams to calculate the number of possible outcomes for compound events. Key topics covered include distinguishing between combinations and permutations, using factorials to find the number of arrangements and groupings, and solving probability problems through listing outcomes and applying counting rules.
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This document discusses permutations and combinations. It provides formulas for calculating the number of permutations and combinations of n objects taken r at a time. Examples are given to demonstrate calculating permutations and combinations to solve problems involving arranging objects in different orders or selecting objects without regard to order. Practice problems are included for students to calculate unknown values and solve word problems involving permutations and combinations.
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This document contains 15 multiple choice questions testing concepts in permutations and combinations. Permutations refer to arrangements of distinguishable objects without repetition, while combinations refer to subsets of objects without regard to order. The document also provides examples and formulas for calculating permutations and combinations, including cases where some objects are identical. It defines key permutation terms like factorial and P(n,r) and illustrates their use through examples like finding the number of ways to arrange 5 people for a portrait.
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Sudheer Mechineni, Head of Application Frameworks, Standard Chartered Bank
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For the full video of this presentation, please visit: https://www.edge-ai-vision.com/2024/06/building-and-scaling-ai-applications-with-the-nx-ai-manager-a-presentation-from-network-optix/
Robin van Emden, Senior Director of Data Science at Network Optix, presents the “Building and Scaling AI Applications with the Nx AI Manager,” tutorial at the May 2024 Embedded Vision Summit.
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We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
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GraphSummit Singapore | Enhancing Changi Airport Group's Passenger Experience...
Dr. Sean Tan, Head of Data Science, Changi Airport Group
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Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
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GraphSummit Singapore | Neo4j Product Vision & Roadmap - Q2 2024
Maruthi Prithivirajan, Head of ASEAN & IN Solution Architecture, Neo4j
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Full-RAG: A modern architecture for hyper-personalization
Mike Del Balso, CEO & Co-Founder at Tecton, presents "Full RAG," a novel approach to AI recommendation systems, aiming to push beyond the limitations of traditional models through a deep integration of contextual insights and real-time data, leveraging the Retrieval-Augmented Generation architecture. This talk will outline Full RAG's potential to significantly enhance personalization, address engineering challenges such as data management and model training, and introduce data enrichment with reranking as a key solution. Attendees will gain crucial insights into the importance of hyperpersonalization in AI, the capabilities of Full RAG for advanced personalization, and strategies for managing complex data integrations for deploying cutting-edge AI solutions.
Artificial Intelligence for XMLDevelopment
In the rapidly evolving landscape of technologies, XML continues to play a vital role in structuring, storing, and transporting data across diverse systems. The recent advancements in artificial intelligence (AI) present new methodologies for enhancing XML development workflows, introducing efficiency, automation, and intelligent capabilities. This presentation will outline the scope and perspective of utilizing AI in XML development. The potential benefits and the possible pitfalls will be highlighted, providing a balanced view of the subject.
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Further emphasis will be placed on the role of AI in developing XSLT, or schemas such as XSD and Schematron. We will address the techniques and strategies adopted to create prompts for generating code, explaining code, or refactoring the code, and the results achieved.
The discussion will extend to how AI can be used to transform XML content. In particular, the focus will be on the use of AI XPath extension functions in XSLT, Schematron, Schematron Quick Fixes, or for XML content refactoring.
The presentation aims to deliver a comprehensive overview of AI usage in XML development, providing attendees with the necessary knowledge to make informed decisions. Whether you’re at the early stages of adopting AI or considering integrating it in advanced XML development, this presentation will cover all levels of expertise.
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Goodbye Windows 11: Make Way for Nitrux Linux 3.5.0!
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12.5 permutations 1
- 1. Lesson 12.5, For use with pages 675-679 1. Choose one of 4 friends, one of 4 restaurants, and one of 3 meeting times. Use the counting principle to find the number of possible choices. You NEED a scientific calculator TODAY! 2. How many seven digit phone numbers are possible if the first digit cannot be a 0?
- 2. Lesson 12.5, For use with pages 675-679 1. Choose one of 4 friends, one of 4 restaurants, and one of 3 meeting times. ANSWER 48 Use the counting principle to find the number of possible choices. You NEED a scientific calculator TODAY! 2. How many seven digit phone numbers are possible if the first digit cannot be a 0? ANSWER 9 x 10 x 10 x 10 x 10 x 10 x 10 = 9,000,000
- 4. Essential Questions • What are the differences between permutations and combinations? • What are the differences between odds and probability? • How is probability used to make predictions? • What are the differences between experimental and theoretical probabilities?
- 5. • When looking at options or choices, we often look at the arrangement, or order, of people, letters, numbers, etc. A permutation is an arrangement in which ORDER is important. Arranging “ABC” is different than “ACB”
- 6. • Let’s look at the three letters – A, B, C • How many different ways can I arrange or order these letters? • What if I add “D” to the list? Now, how many different arrangements?
- 7. • Let’s look at the three letters – A, B, C • How many different ways can I arrange or order these letters? • What if I add “D” to the list? Now, how many different arrangements? 6 ways 24 ways
- 8. EXAMPLE 1 Counting Permutations Music You have five CDs. You can use the counting principle to count the number of permutations of 5 CDs. This is the number of different orders in which you can listen to the CDs. Choices for 1st CD Choices for 2nd CD Choices for 3rd CD Choices for 4th CD Choices for 5th CD 5 4 3 2 1 ANSWER You can listen to the CDs in 120 different orders. = 120
- 9. Swimming Event • There are 8 lanes for the swimming event. If the top 2 swimmers are assigned to lanes 3, and 4 in how many ways can the other 6 swimmers be assigned to their lanes? • (Hint: -- you are only looking at 6 lanes) • For the first of the remaining 6 swimmers, there are 6 possible lanes assignments. Once a selection has been made, there are only 5 lanes left from which to assign the next swimmer. After the second swimmer is assigned, there are only 4 lanes left, and so on. • 6 x 5 x 4 x 3 x 2 x 1 = 720 ways
- 10. • Factorial Notation is used to write the product when the factors are consecutive whole numbers. • 4 factorial is written as 4! and it means 4 x 3 x 2 x 1 or 24 Find the factorial key on your calculator.
- 11. GUIDED PRACTICE for Examples 1 and 2 1. Evaluate 3!. 3! = 3 2 1 2. Evaluate 6!. 6! = 6 5 4 3 2 1 3. Evaluate 1!. 1! = 720 = 6 = 1
- 12. • How many ways can the letters in these words be arranged? MATH PROBABILITY GEOMETRY VOLUME 4! = 24 11! = 39,916,800 8! = 40,320 6! = 720
- 13. Permutations n = total to select from r = the number you are selecting
- 14. EXAMPLE 3 Evaluating a Permutation Poetry Two students are chosen from a group of 6 to read the first and second poems at the school’s poetry reading. To find how many different ways the students can be chosen, find 6P2. 6P2 – == 6! (6 2)! 6! 4! Use permutation formula. = 6 5 4 3 2 1 4 3 2 1 Divide out common factors. = 30 Multiply. ANSWER The students can be chosen in 30 ways. Using the counting principle we would have found it by taking 6 x 5 = 30
- 15. • From the 9 players in the batting order, in how many ways can a baseball manager choose and order the first 3 batters? • 9 x 8 x 7 = 504 or 9P3 = 504 • the first 4 batters? • 3024 ways the first 5 batters? 15,120 ways
- 16. • There are 10 candidates for student government. There are 4 different positions. In how many ways can the student government be chosen? • 10P4 = 5040 ways • Remember: these examples all involve ORDER – arrangements / permutations
- 17. On Line Calculators • Permutations and Combinations – http://www.calctool.org/CALC/math/probability /combinations – http://www.mathsisfun.com/combinatorics/co mbinations-permutations-calculator.html • Factorials – http://joemath.com/math124/Calculator/factori al.htm • www.lewiscentral.k12.ia.us/shipp
- 18. • Assignment: P. 677 #1-21