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Branches of Mathematics
Pure mathematics
Applied mathematics
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Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size (enumerative combinatorics), deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial optimization), and studying combinatorial structures arising in an algebraic context, or applying algebraic techniques to combinatorial problems (algebraic combinatorics).
Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, and combinatorics also has many applications in optimization, computer science, ergodic theory and statistical physics. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is graph theory, which also has numerous natural connections to other areas. Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms.
This page contains text from Wikipedia, the Free Encyclopedia -
http://en.wikipedia.org/wiki/Combinatorics
This article is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License, which means that you can copy and modify it as long as the entire work (including additions) remains under this license.
This article is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License, which means that you can copy and modify it as long as the entire work (including additions) remains under this license.
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Basic Combinatorics- Part 1
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MathHistory29: Combinatorics
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We give a brief historical introduction to the vibrant modern theory of combinatorics, concentrating on examples coming from counting problems, graph theory and generating functions. In particular we look at partitions and Euler's pentagonal theorem, Fibonacci numbers, the Catalan sequence, the Erdos Szekeres theorem, Ramsey theory and the Kirkman Schoolgirls problem. -
Getting Exactly Two Heads (Combinatorics)
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Learn more: http://www.khanacademy.org/video?v=8TIben0bJpU A different way to think about the probability of getting 2 heads in 4 flips. -
Introduction to Combinatorics : Principles of Math
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Subscribe Now: http://www.youtube.com/subscription_center?add_user=Ehow Watch More: http://www.youtube.com/Ehow Combinatorics is a very important course in t... -
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Extremal and Probabilistic Combinatorics Aula 1 Parte 1
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Extremal and Probabilistic Combinatorics In this course we will introduce the student to the basic theorems and proof techniques in extremal graph theory and probabilistic combinatorics. We shall emphasize the close links between these two areas, and provide the background material for modern research fields such as additive combinatorics, monotone and hereditary properties, and graph limits. We also discuss some simple but powerful applications of techniques from functional analysis and linear algebra. The course has no prerequisites. Programa: 1. Ramsey Theory: Finite and infinite versions. Erdös' random proof of the lower bound. Van de -
Die rolling probability | Probability and combinatorics | Precalculus | Khan Academy
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We're thinking about the probability of rolling doubles on a pair of dice. Let's create a grid of all possible outcomes. Watch the next lesson: https://www.khanacademy.org/math/precalculus/prob_comb/independent_events_precalc/v/lebron-asks-about-the-chances-of-making-10-free-throws?utm_source=YT&utm;_medium=Desc&utm;_campaign=Precalculus Missed the previous lesson? https://www.khanacademy.org/math/precalculus/prob_comb/independent_events_precalc/v/getting-at-least-one-heads?utm_source=YT&utm;_medium=Desc&utm;_campaign=Precalculus Precalculus on Khan Academy: You may think that precalculus is simply the course you take before calculus. You wou -
Algebra II: binomial Expansion and Combinatorics
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Unizor - Combinatorics - Permutations
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Combinatorics is a much newer part of mathematics than such classical subjects as Geometry or Algebra. Very important stimulus to its development was the Theory of Probabilities, which is the next subject in this course. Later on, Game Theory and contemporary Computer Science were the other fields of application of the combinatorics. The subject of Permutations is calculating the number of possibilities of putting certain number of objects in certain order. Examples are numerous. For instance, you have to visit 3 different places A, B and C. The order in which you visit them can be ABC, ACB, BAC, BCA, CAB and CBA. Actually, these 6 differen -
Counting and Combinatorics in Discrete Math Part 1
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This is part 1 of learning basic counting and combinations in discrete mathematics. I will give some examples to get you introduced to the idea of finding combinations. -
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Much of enumerative combinatorics concerns the question: "Count the number a_n of elements of a set S_n for n=1,2,..." We discuss four types of answers: an e... -
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How To Do Combinatorics In Poker
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Combinatorics Problems
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- Abstract algebra
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-
Basic Combinatorics- Part 1
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Introduction to combinations and permutations -
MathHistory29: Combinatorics
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We give a brief historical introduction to the vibrant modern theory of combinatorics, concentrating on examples coming from counting problems, graph theory and generating functions. In particular we look at partitions and Euler's pentagonal theorem, Fibonacci numbers, the Catalan sequence, the Erdos Szekeres theorem, Ramsey theory and the Kirkman Schoolgirls problem. -
Getting Exactly Two Heads (Combinatorics)
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Learn more: http://www.khanacademy.org/video?v=8TIben0bJpU A different way to think about the probability of getting 2 heads in 4 flips. -
Introduction to Combinatorics : Principles of Math
Introduction to Combinatorics : Principles of MathIntroduction to Combinatorics : Principles of Math
Subscribe Now: http://www.youtube.com/subscription_center?add_user=Ehow Watch More: http://www.youtube.com/Ehow Combinatorics is a very important course in t... -
Combinatorics with Day[9]: Bijections
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Extremal and Probabilistic Combinatorics Aula 1 Parte 1
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Extremal and Probabilistic Combinatorics In this course we will introduce the student to the basic theorems and proof techniques in extremal graph theory and probabilistic combinatorics. We shall emphasize the close links between these two areas, and provide the background material for modern research fields such as additive combinatorics, monotone and hereditary properties, and graph limits. We also discuss some simple but powerful applications of techniques from functional analysis and linear algebra. The course has no prerequisites. Programa: 1. Ramsey Theory: Finite and infinite versions. Erdös' random proof of the lower bound. Van de -
Die rolling probability | Probability and combinatorics | Precalculus | Khan Academy
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We're thinking about the probability of rolling doubles on a pair of dice. Let's create a grid of all possible outcomes. Watch the next lesson: https://www.khanacademy.org/math/precalculus/prob_comb/independent_events_precalc/v/lebron-asks-about-the-chances-of-making-10-free-throws?utm_source=YT&utm;_medium=Desc&utm;_campaign=Precalculus Missed the previous lesson? https://www.khanacademy.org/math/precalculus/prob_comb/independent_events_precalc/v/getting-at-least-one-heads?utm_source=YT&utm;_medium=Desc&utm;_campaign=Precalculus Precalculus on Khan Academy: You may think that precalculus is simply the course you take before calculus. You wou -
Algebra II: binomial Expansion and Combinatorics
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65 (done another way) - 66, combinatorics and binomial expansions -
Unizor - Combinatorics - Permutations
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Combinatorics is a much newer part of mathematics than such classical subjects as Geometry or Algebra. Very important stimulus to its development was the Theory of Probabilities, which is the next subject in this course. Later on, Game Theory and contemporary Computer Science were the other fields of application of the combinatorics. The subject of Permutations is calculating the number of possibilities of putting certain number of objects in certain order. Examples are numerous. For instance, you have to visit 3 different places A, B and C. The order in which you visit them can be ABC, ACB, BAC, BCA, CAB and CBA. Actually, these 6 differen -
Counting and Combinatorics in Discrete Math Part 1
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This is part 1 of learning basic counting and combinations in discrete mathematics. I will give some examples to get you introduced to the idea of finding combinations. -
Lecture 1 . Enumerative Combinatorics (Federico Ardila)
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Much of enumerative combinatorics concerns the question: "Count the number a_n of elements of a set S_n for n=1,2,..." We discuss four types of answers: an e... -
Alexander Postnikov (MIT) The Combinatorics of the Grassmanian I
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How To Do Combinatorics In Poker
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http://nitreg.com How do you figure out combinatorics in poker? Watch this video and find out. The math is fairly easy and I show you a couple of examples to... -
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Permutations and Combinations 1
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Basic Combinatorics
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I created this video with the YouTube Video Editor (http://www.youtube.com/editor) -
Combinatorics: Venn Diagrams and the Inclusion-Exclusion Principle
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Probability using Combinatorics
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GMAT Online Math - Permutation & Combinatorics 1 | Manhattan Review GMAT Prep
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http://www.manhattanreview.com/gmat-online/. This video covers the permutation and combinatorics problems in the GMAT Math Problem Solving section. Manhattan... -
Unizor - Combinatorics - Advanced Problems 1.1.
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Problem 1 What is the number of partial permutations of N given objects by K objects under a restriction that each such partial permutation must contain X pa... -
Combination (Combinatorics & Probability) Cards Word Problem #1
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For many similar practice questions and video explanations, visit us at: http://www.acemymathcourse.com. -
Why Blockers Matter...Intro to Combinatorics | Poker Tips
Why Blockers Matter...Intro to Combinatorics | Poker TipsWhy Blockers Matter...Intro to Combinatorics | Poker Tips
Got a Question? write me! evan@gripsed.com Want to Support the Gripsed Project Training Project? Click Show More to find out How :) Sign Up to our Sponsor Sites and Make a Deposit! Click Here: http://888poker.com/promotions/?sr=1062353 Or Here: http://online.titanpoker.com/promoRedirect?key=ej0xNDI5OTY4NyZsPTEzNTIyMzExJnA9OTU0NTAx Donate via Paypal: https://www.paypal.com/cgi-bin/webscr?cmd=_s-xclick&hosted;_button_id=D8TMLEW4V27WJ Donate Via Pokerstars: "SenorPokes" Thanks for Taking the Time to Tune in! And Thanks for Your Support!
Basic Combinatorics- Part 1
- Order: Reorder
- Duration: 13:15
- Updated: 26 Apr 2013
Introduction to combinations and permutations
http://wn.com/Basic_Combinatorics-_Part_1
Introduction to combinations and permutations
- published: 26 Apr 2013
- views: 7529
MathHistory29: Combinatorics
- Order: Reorder
- Duration: 41:01
- Updated: 03 Jun 2015
We give a brief historical introduction to the vibrant modern theory of combinatorics, concentrating on examples coming from counting problems, graph theory and generating functions. In particular we look at partitions and Euler's pentagonal theorem, Fibonacci numbers, the Catalan sequence, the Erdos Szekeres theorem, Ramsey theory and the Kirkman Schoolgirls problem.
http://wn.com/MathHistory29_Combinatorics
We give a brief historical introduction to the vibrant modern theory of combinatorics, concentrating on examples coming from counting problems, graph theory and generating functions. In particular we look at partitions and Euler's pentagonal theorem, Fibonacci numbers, the Catalan sequence, the Erdos Szekeres theorem, Ramsey theory and the Kirkman Schoolgirls problem.
- published: 03 Jun 2015
- views: 115
Getting Exactly Two Heads (Combinatorics)
- Order: Reorder
- Duration: 10:00
- Updated: 11 Aug 2013
- author: khanacademy
Learn more: http://www.khanacademy.org/video?v=8TIben0bJpU A different way to think about the probability of getting 2 heads in 4 flips.
http://wn.com/Getting_Exactly_Two_Heads_(Combinatorics)
Learn more: http://www.khanacademy.org/video?v=8TIben0bJpU A different way to think about the probability of getting 2 heads in 4 flips.
- published: 10 Aug 2011
- views: 66399
- author: khanacademy
Introduction to Combinatorics : Principles of Math
Subscribe Now: http://www.youtube.com/subscription_center?add_user=Ehow Watch More: http://www.youtube.com/Ehow Combinatorics is a very important course in t...
http://wn.com/Introduction_to_Combinatorics_Principles_of_Math
Subscribe Now: http://www.youtube.com/subscription_center?add_user=Ehow Watch More: http://www.youtube.com/Ehow Combinatorics is a very important course in t...
- published: 16 Nov 2012
- views: 4133
- author: eHow
![Combinatorics with Day[9]: Bijections](http://web.archive.org./web/20150818163115im_/http://i.ytimg.com/vi/rJmlNIKMVgA/0.jpg "Combinatorics with Day[9]: Bijections")
Combinatorics with Day[9]: Bijections
Combinatorics with Day[9]: Bijections.
http://wn.com/Combinatorics_with_Day_9_Bijections
Extremal and Probabilistic Combinatorics Aula 1 Parte 1
- Order: Reorder
- Duration: 47:42
- Updated: 03 Feb 2015
Extremal and Probabilistic Combinatorics
In this course we will introduce the student to the basic theorems and proof techniques in extremal graph theory and probabilistic combinatorics. We shall emphasize the close links between these two areas, and provide the background material for modern research fields such as additive combinatorics, monotone and hereditary properties, and graph limits. We also discuss some simple but powerful applications of techniques from functional analysis and linear algebra. The course has no prerequisites.
Programa:
1. Ramsey Theory: Finite and infinite versions. Erdös' random proof of the lower bound. Van der Waarden's Theorem. Statement of Szemerédi's Theorem.
2. Extremal Graph Theory: The theorems of Turán, Erdös-Stone and Kovari-Sós-Turán.
3. The Erdös-Renyi Random Graph: Graphs with high girth and chromatic number. Extremal number of C2k. 1st and 2nd moment methods. Janson's inequality. The giant component.
4. Analytic and Algebraic Methods: The Kneser graph and the Borsuk-Ulam Theorem. The Frankl-Wilson inequality and Borsuk's Conjecture.
5. The Szemerédi Regularity Lemma: Statement and applications, e.g., proof of Erdös-Stone, Erdös-Frankl-Rödl. Proof of Roth's Theorem via the triangle-removal lemma.
6. Dependent Random Choice: Applications, including the proof of the Balog-Szemerédi-Gowers Theorem.
Robert Morris
http://wn.com/Extremal_and_Probabilistic_Combinatorics_Aula_1_Parte_1
Extremal and Probabilistic Combinatorics
In this course we will introduce the student to the basic theorems and proof techniques in extremal graph theory and probabilistic combinatorics. We shall emphasize the close links between these two areas, and provide the background material for modern research fields such as additive combinatorics, monotone and hereditary properties, and graph limits. We also discuss some simple but powerful applications of techniques from functional analysis and linear algebra. The course has no prerequisites.
Programa:
1. Ramsey Theory: Finite and infinite versions. Erdös' random proof of the lower bound. Van der Waarden's Theorem. Statement of Szemerédi's Theorem.
2. Extremal Graph Theory: The theorems of Turán, Erdös-Stone and Kovari-Sós-Turán.
3. The Erdös-Renyi Random Graph: Graphs with high girth and chromatic number. Extremal number of C2k. 1st and 2nd moment methods. Janson's inequality. The giant component.
4. Analytic and Algebraic Methods: The Kneser graph and the Borsuk-Ulam Theorem. The Frankl-Wilson inequality and Borsuk's Conjecture.
5. The Szemerédi Regularity Lemma: Statement and applications, e.g., proof of Erdös-Stone, Erdös-Frankl-Rödl. Proof of Roth's Theorem via the triangle-removal lemma.
6. Dependent Random Choice: Applications, including the proof of the Balog-Szemerédi-Gowers Theorem.
Robert Morris
- published: 03 Feb 2015
- views: 3
Die rolling probability | Probability and combinatorics | Precalculus | Khan Academy
- Order: Reorder
- Duration: 5:15
- Updated: 13 Jul 2015
We're thinking about the probability of rolling doubles on a pair of dice. Let's create a grid of all possible outcomes.
Watch the next lesson: https://www.khanacademy.org/math/precalculus/prob_comb/independent_events_precalc/v/lebron-asks-about-the-chances-of-making-10-free-throws?utm_source=YT&utm;_medium=Desc&utm;_campaign=Precalculus
Missed the previous lesson?
https://www.khanacademy.org/math/precalculus/prob_comb/independent_events_precalc/v/getting-at-least-one-heads?utm_source=YT&utm;_medium=Desc&utm;_campaign=Precalculus
Precalculus on Khan Academy: You may think that precalculus is simply the course you take before calculus. You would be right, of course, but that definition doesn't mean anything unless you have some knowledge of what calculus is. Let's keep it simple, shall we? Calculus is a conceptual framework which provides systematic techniques for solving problems. These problems are appropriately applicable to analytic geometry and algebra. Therefore....precalculus gives you the background for the mathematical concepts, problems, issues and techniques that appear in calculus, including trigonometry, functions, complex numbers, vectors, matrices, and others. There you have it ladies and gentlemen....an introduction to precalculus!
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to Khan Academy’s Precalculus channel:
https://www.youtube.com/channel/UCBeHztHRWuVvnlwm20u2hNA?sub_confirmation=1
Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy
http://wn.com/Die_rolling_probability_|_Probability_and_combinatorics_|_Precalculus_|_Khan_Academy
We're thinking about the probability of rolling doubles on a pair of dice. Let's create a grid of all possible outcomes.
Watch the next lesson: https://www.khanacademy.org/math/precalculus/prob_comb/independent_events_precalc/v/lebron-asks-about-the-chances-of-making-10-free-throws?utm_source=YT&utm;_medium=Desc&utm;_campaign=Precalculus
Missed the previous lesson?
https://www.khanacademy.org/math/precalculus/prob_comb/independent_events_precalc/v/getting-at-least-one-heads?utm_source=YT&utm;_medium=Desc&utm;_campaign=Precalculus
Precalculus on Khan Academy: You may think that precalculus is simply the course you take before calculus. You would be right, of course, but that definition doesn't mean anything unless you have some knowledge of what calculus is. Let's keep it simple, shall we? Calculus is a conceptual framework which provides systematic techniques for solving problems. These problems are appropriately applicable to analytic geometry and algebra. Therefore....precalculus gives you the background for the mathematical concepts, problems, issues and techniques that appear in calculus, including trigonometry, functions, complex numbers, vectors, matrices, and others. There you have it ladies and gentlemen....an introduction to precalculus!
About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.
For free. For everyone. Forever. #YouCanLearnAnything
Subscribe to Khan Academy’s Precalculus channel:
https://www.youtube.com/channel/UCBeHztHRWuVvnlwm20u2hNA?sub_confirmation=1
Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy
- published: 13 Jul 2015
- views: 89446
Algebra II: binomial Expansion and Combinatorics
- Order: Reorder
- Duration: 12:49
- Updated: 24 Dec 2008
65 (done another way) - 66, combinatorics and binomial expansions
http://wn.com/Algebra_II_binomial_Expansion_and_Combinatorics
65 (done another way) - 66, combinatorics and binomial expansions
- published: 24 Dec 2008
- views: 44118
Unizor - Combinatorics - Permutations
- Order: Reorder
- Duration: 18:10
- Updated: 12 May 2014
Combinatorics is a much newer part of mathematics than such classical subjects as Geometry or Algebra. Very important stimulus to its development was the Theory of Probabilities, which is the next subject in this course. Later on, Game Theory and contemporary Computer Science were the other fields of application of the combinatorics.
The subject of Permutations is calculating the number of possibilities of putting certain number of objects in certain order. Examples are numerous. For instance, you have to visit 3 different places A, B and C. The order in which you visit them can be ABC, ACB, BAC, BCA, CAB and CBA. Actually, these 6 different ways to visit 3 places are the only possible ones, there are no other ways of putting them in the order of visiting. In many cases it's very important to know how many possibilities to do something like this visiting exists. That's the task of calculating the permutations.
The simplest form of permutations is putting N objects in some order. For example, we have numbers from 1 to N and want to write them down in some order. We can write them in ascending order, in descending order or in many other types of order. But what is the number of different ways we can write them down?
Ordering of N objects assumes that we have to choose the object #1, then, among other objects we have to choose the object #2, then among whatever is left we have to choose the object #3 etc. Continuing this process to the end, we finally get to object #N.
For #1 object we have N candidates. With each of them there remain N−1 candidates for #2 spot. So, we have N·(N−1) choices for the first 2 places. With each of them there remain N−2 candidates for #3 spot, which brings the total number of variants for the first 3 places to N·(N−1)·(N−2). Continuing this process to all N places, we come up with the total number of variants to position N objects in some order equal to N·(N−1)·(N−2)·...·2·1.
This quantity is usually written in the form N! and pronounced "N factorial".
At this point we'd like to suggest certain criticism to many textbooks that explain this more or less in the same way. The explanation above is just the explanation, not a proof. So, to stop at this point, as many textbooks do, cannot be considered a truly mathematical approach. We have to prove it, and that's exactly what we are going to do next.
Proof
We are going to prove by induction that the number of permutations of N different objects equals to
N·(N−1)·(N−2)·...·2·1 = N!
(that is, N factorial)
Induction step 1. The formula is obviously correct for N=1 because it produces the result 1 and there is only one way to position a single object in some order.
Induction step 2. Assume that the formula is correct for N=K, that is that the number of permutations of K objects is K!.
Induction step 3. Consider we have N=K+1 objects. Each permutation (the way of ordering) of these objects involves positioning of the first K objects in some way (and there are K! of these ways, as we assumed on step 2) and placing the (K+1)th object somewhere among these first K objects. The (K+1)th object can stand before the first, between the first and the second, between the second and the third,..., between (K−1)th and Kth objects and, finally, after the Kth object. This constitutes K+1 different positions for the (K+1)th object. Therefore, for any permutation of the first K objects there are K+1 different permutations of the whole set of K+1 objects. Hence, the total number of permutations of K+1 objects equals to K! multiplied by (K+1).
But K! · (K+1) = (K+1)!, as follows from the definition of the "factorial" as a product of all natural numbers from 1 to a given number. That proves that the formula retains its form when we move from N=K objects to N=K+1 objects.
This completes the proof.
We will use the symbol P(N) to designate the number of permutations of N objects. We, therefore, have proved that
P(N) = N!
http://wn.com/Unizor_-_Combinatorics_-_Permutations
Combinatorics is a much newer part of mathematics than such classical subjects as Geometry or Algebra. Very important stimulus to its development was the Theory of Probabilities, which is the next subject in this course. Later on, Game Theory and contemporary Computer Science were the other fields of application of the combinatorics.
The subject of Permutations is calculating the number of possibilities of putting certain number of objects in certain order. Examples are numerous. For instance, you have to visit 3 different places A, B and C. The order in which you visit them can be ABC, ACB, BAC, BCA, CAB and CBA. Actually, these 6 different ways to visit 3 places are the only possible ones, there are no other ways of putting them in the order of visiting. In many cases it's very important to know how many possibilities to do something like this visiting exists. That's the task of calculating the permutations.
The simplest form of permutations is putting N objects in some order. For example, we have numbers from 1 to N and want to write them down in some order. We can write them in ascending order, in descending order or in many other types of order. But what is the number of different ways we can write them down?
Ordering of N objects assumes that we have to choose the object #1, then, among other objects we have to choose the object #2, then among whatever is left we have to choose the object #3 etc. Continuing this process to the end, we finally get to object #N.
For #1 object we have N candidates. With each of them there remain N−1 candidates for #2 spot. So, we have N·(N−1) choices for the first 2 places. With each of them there remain N−2 candidates for #3 spot, which brings the total number of variants for the first 3 places to N·(N−1)·(N−2). Continuing this process to all N places, we come up with the total number of variants to position N objects in some order equal to N·(N−1)·(N−2)·...·2·1.
This quantity is usually written in the form N! and pronounced "N factorial".
At this point we'd like to suggest certain criticism to many textbooks that explain this more or less in the same way. The explanation above is just the explanation, not a proof. So, to stop at this point, as many textbooks do, cannot be considered a truly mathematical approach. We have to prove it, and that's exactly what we are going to do next.
Proof
We are going to prove by induction that the number of permutations of N different objects equals to
N·(N−1)·(N−2)·...·2·1 = N!
(that is, N factorial)
Induction step 1. The formula is obviously correct for N=1 because it produces the result 1 and there is only one way to position a single object in some order.
Induction step 2. Assume that the formula is correct for N=K, that is that the number of permutations of K objects is K!.
Induction step 3. Consider we have N=K+1 objects. Each permutation (the way of ordering) of these objects involves positioning of the first K objects in some way (and there are K! of these ways, as we assumed on step 2) and placing the (K+1)th object somewhere among these first K objects. The (K+1)th object can stand before the first, between the first and the second, between the second and the third,..., between (K−1)th and Kth objects and, finally, after the Kth object. This constitutes K+1 different positions for the (K+1)th object. Therefore, for any permutation of the first K objects there are K+1 different permutations of the whole set of K+1 objects. Hence, the total number of permutations of K+1 objects equals to K! multiplied by (K+1).
But K! · (K+1) = (K+1)!, as follows from the definition of the "factorial" as a product of all natural numbers from 1 to a given number. That proves that the formula retains its form when we move from N=K objects to N=K+1 objects.
This completes the proof.
We will use the symbol P(N) to designate the number of permutations of N objects. We, therefore, have proved that
P(N) = N!
- published: 12 May 2014
- views: 1
Counting and Combinatorics in Discrete Math Part 1
- Order: Reorder
- Duration: 10:23
- Updated: 01 Dec 2014
This is part 1 of learning basic counting and combinations in discrete mathematics. I will give some examples to get you introduced to the idea of finding combinations.
http://wn.com/Counting_and_Combinatorics_in_Discrete_Math_Part_1
This is part 1 of learning basic counting and combinations in discrete mathematics. I will give some examples to get you introduced to the idea of finding combinations.
- published: 01 Dec 2014
- views: 7
Lecture 1 . Enumerative Combinatorics (Federico Ardila)
- Order: Reorder
- Duration: 68:24
- Updated: 18 Aug 2014
- author: Federico Ardila
Much of enumerative combinatorics concerns the question: "Count the number a_n of elements of a set S_n for n=1,2,..." We discuss four types of answers: an e...
http://wn.com/Lecture_1_._Enumerative_Combinatorics_(Federico_Ardila)
Much of enumerative combinatorics concerns the question: "Count the number a_n of elements of a set S_n for n=1,2,..." We discuss four types of answers: an e...
- published: 30 Aug 2013
- views: 3409
- author: Federico Ardila
Alexander Postnikov (MIT) The Combinatorics of the Grassmanian I
- Order: Reorder
- Duration: 89:36
- Updated: 22 Jan 2015
Alexander Postnikov (MIT)
The Combinatorics of the Grassmanian I
http://wn.com/Alexander_Postnikov_(MIT)_The_Combinatorics_of_the_Grassmanian_I
Alexander Postnikov (MIT)
The Combinatorics of the Grassmanian I
- published: 22 Jan 2015
- views: 40
How To Do Combinatorics In Poker
- Order: Reorder
- Duration: 20:31
- Updated: 23 Aug 2014
- author: nitregpoker
http://nitreg.com How do you figure out combinatorics in poker? Watch this video and find out. The math is fairly easy and I show you a couple of examples to...
http://wn.com/How_To_Do_Combinatorics_In_Poker
http://nitreg.com How do you figure out combinatorics in poker? Watch this video and find out. The math is fairly easy and I show you a couple of examples to...
- published: 12 Nov 2012
- views: 2835
- author: nitregpoker
Combinatorics Problems
- Order: Reorder
- Duration: 8:28
- Updated: 16 Jan 2013
- author: Catherine Boersma
http://wn.com/Combinatorics_Problems
Permutations and Combinations 1
- Order: Reorder
- Duration: 3:45
- Updated: 05 Sep 2014
- author: Khan Academy
U12_L2_T3_we1 Permutations and Combinations 1 More free lessons at: http://www.khanacademy.org/video?v=oQpKtm5TtxU Content provided by TheNROCproject.org - (...
http://wn.com/Permutations_and_Combinations_1
U12_L2_T3_we1 Permutations and Combinations 1 More free lessons at: http://www.khanacademy.org/video?v=oQpKtm5TtxU Content provided by TheNROCproject.org - (...
- published: 27 Jun 2010
- views: 255892
- author: Khan Academy
0.4 Analytic Combinatorics 2915)
- Order: Reorder
- Duration: 29:16
- Updated: 27 Apr 2014
- author: Wenlong ZHAO
http://wn.com/0.4_Analytic_Combinatorics_2915)
Basic Combinatorics
I created this video with the YouTube Video Editor (http://www.youtube.com/editor)
http://wn.com/Basic_Combinatorics
I created this video with the YouTube Video Editor (http://www.youtube.com/editor)
- published: 25 Nov 2013
- views: 525
- author: MrCHSMath
Combinatorics: Venn Diagrams and the Inclusion-Exclusion Principle
- Order: Reorder
- Duration: 13:12
- Updated: 30 May 2014
- author: Paul Griffin
A look at Venn Diagrams and the Inclusion-Exclusion Principle. Includes the solution to a question from Richard G. Brown's "Advanced Mathematics: Precalculus...
http://wn.com/Combinatorics_Venn_Diagrams_and_the_Inclusion-Exclusion_Principle
A look at Venn Diagrams and the Inclusion-Exclusion Principle. Includes the solution to a question from Richard G. Brown's "Advanced Mathematics: Precalculus...
- published: 10 Mar 2013
- views: 1672
- author: Paul Griffin
GMAT Online Math - Permutation & Combinatorics 1 | Manhattan Review GMAT Prep
- Order: Reorder
- Duration: 6:26
- Updated: 07 Aug 2014
- author: ManhattanReview
http://www.manhattanreview.com/gmat-online/. This video covers the permutation and combinatorics problems in the GMAT Math Problem Solving section. Manhattan...
http://wn.com/GMAT_Online_Math_-_Permutation_&_Combinatorics_1_|_Manhattan_Review_GMAT_Prep
http://www.manhattanreview.com/gmat-online/. This video covers the permutation and combinatorics problems in the GMAT Math Problem Solving section. Manhattan...
- published: 28 Dec 2009
- views: 7009
- author: ManhattanReview
Unizor - Combinatorics - Advanced Problems 1.1.
- Order: Reorder
- Duration: 21:22
- Updated: 11 Aug 2014
- author: Zor Shekhtman
Problem 1 What is the number of partial permutations of N given objects by K objects under a restriction that each such partial permutation must contain X pa...
http://wn.com/Unizor_-_Combinatorics_-_Advanced_Problems_1.1.
Problem 1 What is the number of partial permutations of N given objects by K objects under a restriction that each such partial permutation must contain X pa...
- published: 24 May 2014
- views: 52
- author: Zor Shekhtman
Combination (Combinatorics & Probability) Cards Word Problem #1
- Order: Reorder
- Duration: 4:31
- Updated: 31 Jul 2014
- author: AcademicLeadersEd
For many similar practice questions and video explanations, visit us at: http://www.acemymathcourse.com.
http://wn.com/Combination_(Combinatorics_&_Probability)_Cards_Word_Problem_1
For many similar practice questions and video explanations, visit us at: http://www.acemymathcourse.com.
- published: 14 Feb 2013
- views: 1075
- author: AcademicLeadersEd
Why Blockers Matter...Intro to Combinatorics | Poker Tips
- Order: Reorder
- Duration: 14:49
- Updated: 09 Dec 2013
Got a Question? write me! evan@gripsed.com
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- published: 09 Dec 2013
- views: 1793
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41:01
MathHistory29: Combinatorics
We give a brief historical introduction to the vibrant modern theory of combinatorics, con...
published: 03 Jun 2015
MathHistory29: Combinatorics
MathHistory29: Combinatorics
We give a brief historical introduction to the vibrant modern theory of combinatorics, concentrating on examples coming from counting problems, graph theory and generating functions. In particular we look at partitions and Euler's pentagonal theorem, Fibonacci numbers, the Catalan sequence, the Erdos Szekeres theorem, Ramsey theory and the Kirkman Schoolgirls problem.- published: 03 Jun 2015
- views: 115
10:00
Getting Exactly Two Heads (Combinatorics)
Learn more: http://www.khanacademy.org/video?v=8TIben0bJpU A different way to think about ...
published: 10 Aug 2011
author: khanacademy
Getting Exactly Two Heads (Combinatorics)
Getting Exactly Two Heads (Combinatorics)
Learn more: http://www.khanacademy.org/video?v=8TIben0bJpU A different way to think about the probability of getting 2 heads in 4 flips.- published: 10 Aug 2011
- views: 66399
- author: khanacademy
1:38
Introduction to Combinatorics : Principles of Math
Subscribe Now: http://www.youtube.com/subscription_center?add_user=Ehow Watch More: http:/...
published: 16 Nov 2012
author: eHow
Introduction to Combinatorics : Principles of Math
Introduction to Combinatorics : Principles of Math
Subscribe Now: http://www.youtube.com/subscription_center?add_user=Ehow Watch More: http://www.youtube.com/Ehow Combinatorics is a very important course in t...- published: 16 Nov 2012
- views: 4133
- author: eHow
47:42
Extremal and Probabilistic Combinatorics Aula 1 Parte 1
Extremal and Probabilistic Combinatorics
In this course we will introduce the student to ...
published: 03 Feb 2015
Extremal and Probabilistic Combinatorics Aula 1 Parte 1
Extremal and Probabilistic Combinatorics Aula 1 Parte 1
Extremal and Probabilistic Combinatorics In this course we will introduce the student to the basic theorems and proof techniques in extremal graph theory and probabilistic combinatorics. We shall emphasize the close links between these two areas, and provide the background material for modern research fields such as additive combinatorics, monotone and hereditary properties, and graph limits. We also discuss some simple but powerful applications of techniques from functional analysis and linear algebra. The course has no prerequisites. Programa: 1. Ramsey Theory: Finite and infinite versions. Erdös' random proof of the lower bound. Van der Waarden's Theorem. Statement of Szemerédi's Theorem. 2. Extremal Graph Theory: The theorems of Turán, Erdös-Stone and Kovari-Sós-Turán. 3. The Erdös-Renyi Random Graph: Graphs with high girth and chromatic number. Extremal number of C2k. 1st and 2nd moment methods. Janson's inequality. The giant component. 4. Analytic and Algebraic Methods: The Kneser graph and the Borsuk-Ulam Theorem. The Frankl-Wilson inequality and Borsuk's Conjecture. 5. The Szemerédi Regularity Lemma: Statement and applications, e.g., proof of Erdös-Stone, Erdös-Frankl-Rödl. Proof of Roth's Theorem via the triangle-removal lemma. 6. Dependent Random Choice: Applications, including the proof of the Balog-Szemerédi-Gowers Theorem. Robert Morris- published: 03 Feb 2015
- views: 3
5:15
Die rolling probability | Probability and combinatorics | Precalculus | Khan Academy
We're thinking about the probability of rolling doubles on a pair of dice. Let's create a ...
published: 13 Jul 2015
Die rolling probability | Probability and combinatorics | Precalculus | Khan Academy
Die rolling probability | Probability and combinatorics | Precalculus | Khan Academy
We're thinking about the probability of rolling doubles on a pair of dice. Let's create a grid of all possible outcomes. Watch the next lesson: https://www.khanacademy.org/math/precalculus/prob_comb/independent_events_precalc/v/lebron-asks-about-the-chances-of-making-10-free-throws?utm_source=YT&utm;_medium=Desc&utm;_campaign=Precalculus Missed the previous lesson? https://www.khanacademy.org/math/precalculus/prob_comb/independent_events_precalc/v/getting-at-least-one-heads?utm_source=YT&utm;_medium=Desc&utm;_campaign=Precalculus Precalculus on Khan Academy: You may think that precalculus is simply the course you take before calculus. You would be right, of course, but that definition doesn't mean anything unless you have some knowledge of what calculus is. Let's keep it simple, shall we? Calculus is a conceptual framework which provides systematic techniques for solving problems. These problems are appropriately applicable to analytic geometry and algebra. Therefore....precalculus gives you the background for the mathematical concepts, problems, issues and techniques that appear in calculus, including trigonometry, functions, complex numbers, vectors, matrices, and others. There you have it ladies and gentlemen....an introduction to precalculus! About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We've also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content. For free. For everyone. Forever. #YouCanLearnAnything Subscribe to Khan Academy’s Precalculus channel: https://www.youtube.com/channel/UCBeHztHRWuVvnlwm20u2hNA?sub_confirmation=1 Subscribe to Khan Academy: https://www.youtube.com/subscription_center?add_user=khanacademy- published: 13 Jul 2015
- views: 89446
12:49
Algebra II: binomial Expansion and Combinatorics
65 (done another way) - 66, combinatorics and binomial expansions...
published: 24 Dec 2008
Algebra II: binomial Expansion and Combinatorics
Algebra II: binomial Expansion and Combinatorics
65 (done another way) - 66, combinatorics and binomial expansions- published: 24 Dec 2008
- views: 44118
18:10
Unizor - Combinatorics - Permutations
Combinatorics is a much newer part of mathematics than such classical subjects as Geometry...
published: 12 May 2014
Unizor - Combinatorics - Permutations
Unizor - Combinatorics - Permutations
Combinatorics is a much newer part of mathematics than such classical subjects as Geometry or Algebra. Very important stimulus to its development was the Theory of Probabilities, which is the next subject in this course. Later on, Game Theory and contemporary Computer Science were the other fields of application of the combinatorics. The subject of Permutations is calculating the number of possibilities of putting certain number of objects in certain order. Examples are numerous. For instance, you have to visit 3 different places A, B and C. The order in which you visit them can be ABC, ACB, BAC, BCA, CAB and CBA. Actually, these 6 different ways to visit 3 places are the only possible ones, there are no other ways of putting them in the order of visiting. In many cases it's very important to know how many possibilities to do something like this visiting exists. That's the task of calculating the permutations. The simplest form of permutations is putting N objects in some order. For example, we have numbers from 1 to N and want to write them down in some order. We can write them in ascending order, in descending order or in many other types of order. But what is the number of different ways we can write them down? Ordering of N objects assumes that we have to choose the object #1, then, among other objects we have to choose the object #2, then among whatever is left we have to choose the object #3 etc. Continuing this process to the end, we finally get to object #N. For #1 object we have N candidates. With each of them there remain N−1 candidates for #2 spot. So, we have N·(N−1) choices for the first 2 places. With each of them there remain N−2 candidates for #3 spot, which brings the total number of variants for the first 3 places to N·(N−1)·(N−2). Continuing this process to all N places, we come up with the total number of variants to position N objects in some order equal to N·(N−1)·(N−2)·...·2·1. This quantity is usually written in the form N! and pronounced "N factorial". At this point we'd like to suggest certain criticism to many textbooks that explain this more or less in the same way. The explanation above is just the explanation, not a proof. So, to stop at this point, as many textbooks do, cannot be considered a truly mathematical approach. We have to prove it, and that's exactly what we are going to do next. Proof We are going to prove by induction that the number of permutations of N different objects equals to N·(N−1)·(N−2)·...·2·1 = N! (that is, N factorial) Induction step 1. The formula is obviously correct for N=1 because it produces the result 1 and there is only one way to position a single object in some order. Induction step 2. Assume that the formula is correct for N=K, that is that the number of permutations of K objects is K!. Induction step 3. Consider we have N=K+1 objects. Each permutation (the way of ordering) of these objects involves positioning of the first K objects in some way (and there are K! of these ways, as we assumed on step 2) and placing the (K+1)th object somewhere among these first K objects. The (K+1)th object can stand before the first, between the first and the second, between the second and the third,..., between (K−1)th and Kth objects and, finally, after the Kth object. This constitutes K+1 different positions for the (K+1)th object. Therefore, for any permutation of the first K objects there are K+1 different permutations of the whole set of K+1 objects. Hence, the total number of permutations of K+1 objects equals to K! multiplied by (K+1). But K! · (K+1) = (K+1)!, as follows from the definition of the "factorial" as a product of all natural numbers from 1 to a given number. That proves that the formula retains its form when we move from N=K objects to N=K+1 objects. This completes the proof. We will use the symbol P(N) to designate the number of permutations of N objects. We, therefore, have proved that P(N) = N!- published: 12 May 2014
- views: 1
10:23
Counting and Combinatorics in Discrete Math Part 1
This is part 1 of learning basic counting and combinations in discrete mathematics. I will...
published: 01 Dec 2014
Counting and Combinatorics in Discrete Math Part 1
Counting and Combinatorics in Discrete Math Part 1
This is part 1 of learning basic counting and combinations in discrete mathematics. I will give some examples to get you introduced to the idea of finding combinations.- published: 01 Dec 2014
- views: 7
68:24
Lecture 1 . Enumerative Combinatorics (Federico Ardila)
Much of enumerative combinatorics concerns the question: "Count the number a_n of elements...
published: 30 Aug 2013
author: Federico Ardila
Lecture 1 . Enumerative Combinatorics (Federico Ardila)
Lecture 1 . Enumerative Combinatorics (Federico Ardila)
Much of enumerative combinatorics concerns the question: "Count the number a_n of elements of a set S_n for n=1,2,..." We discuss four types of answers: an e...- published: 30 Aug 2013
- views: 3409
- author: Federico Ardila
89:36
Alexander Postnikov (MIT) The Combinatorics of the Grassmanian I
Alexander Postnikov (MIT)
The Combinatorics of the Grassmanian I...
published: 22 Jan 2015
Alexander Postnikov (MIT) The Combinatorics of the Grassmanian I
Alexander Postnikov (MIT) The Combinatorics of the Grassmanian I
Alexander Postnikov (MIT) The Combinatorics of the Grassmanian I- published: 22 Jan 2015
- views: 40
20:31
How To Do Combinatorics In Poker
http://nitreg.com How do you figure out combinatorics in poker? Watch this video and find ...
published: 12 Nov 2012
author: nitregpoker
How To Do Combinatorics In Poker
How To Do Combinatorics In Poker
http://nitreg.com How do you figure out combinatorics in poker? Watch this video and find out. The math is fairly easy and I show you a couple of examples to...- published: 12 Nov 2012
- views: 2835
- author: nitregpoker
8:28
Combinatorics Problems
...
published: 16 Jan 2013
author: Catherine Boersma
Combinatorics Problems
photo: NASA
![The Space Shuttle Discovery approaches the International Space Station for docking but before the link-up occurred, the orbiter "posed" for a thorough series of inspection photos. Discovery docked at the station´s Pressurized Mating Adapt](http://web.archive.org./web/20150818163115im_/http://cdn6.wn.com/ph/img/67/09/315d2c5dabc666421822fbbe2fe9-small.jpg "The Space Shuttle Discovery approaches the International Space Station for docking but before the link-up occurred, the orbiter "posed" for a thorough series of inspection photos. Discovery docked at the station´s Pressurized Mating Adapt")
17 Aug 2015
Canadian space firm granted the US patent for an elevator designed to take astronauts up into the stratosphere, so they can then be propelled into space. @mahitagajanan. email. A Canadian space firm is one step closer to revolutionizing space travel with a simple idea – instead of taking a rocket ship, why not take a giant elevator into space? ... “Astronauts would ascend to 20 km by electrical elevator ... ....
photo: Creative Commons / Jane023
17 Aug 2015
This nightmare is all too real. . An Australian couple awoke shortly before 2 a.m. on Monday to discover a naked stranger snoozing alongside them in bed, The Daily Telegraph reports. Katie and Chris, of Maroubra, believe that the uninvited 25-year-old visitor entered through a window left open for their cat. "Chris is on one side, I'm in the middle and then our stranger on the end right in with us," Katie told reporters ... ....
photo: WN / Imran Nissar
17 Aug 2015
Whether sitting on a train or having dinner at a restaurant, many people find it hard to stop fiddling with their mobile phones – firing off a never-ending stream of Facebook, Instagram and Twitter posts ... New research finds that the most frequent mobile phone and internet users are the most likely to be distracted, for example by being prone to missing important appointments and daydreaming while having a conversation ... --> ... ....
photo: AP / Sackchai Lalit
18 Aug 2015
Thailand's junta leader said Tuesday security forces had identified a suspect in an unprecedented Bangkok bombing that targeted foreigners at a packed religious shrine, killing at least 21 people. The attack occurred at dusk on Monday in one of the Thai capital's most popular tourism hubs, ripping through a crowd of worshippers at the Hindu shrine close to five-star hotels and upscale shopping malls ... "Today there is a suspect......
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18 Aug 2015
(Source. Penn State Beaver) Penn State Beaver faculty publications are announced. 8/17/2015 - ... Dr. Zhongyuan Che, associate professor of mathematics, coauthored "Sharp bounds on the size of pairable graphs and pairable bipartite graphs," which can be found online in the Australasian Journal of Combinatorics., Volume 62(2) (2015 ... LXXII, No ... Her research interests include graph theory and combinatorics as well as bioinformatics....
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Public Technologies
17 Aug 2015
(Source. University of Arkansas) Aug. 17, 2015. FAYETTEVILLE, Ark. - Paolo Mantero will join the Department of Mathematical Sciences this fall as an assistant professor in the J. William Fulbright College of Arts and Sciences ... Mantero's primary research focus is commutative algebra and its interactions with algebraic geometry, combinatorics and homological algebra ... "The department is thrilled to welcome Dr ... Contacts ... J ... J ... distributed by....
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Gifted and talented students converge on campus for 'BRAINways Academicus' (Central Queensland University)
Public Technologies04 Aug 2015
(Source. Central Queensland University). Published.04 August 2015. CQUniversity Associate VC (Rockhampton Region) Kim Harrington welcomes the BRAINways Academicus event participants ... BRAINways EDUCATION is an independent, non-funded organisation, whose programs provide learning opportunities for highly able and gifted students which are tailored for their special educational needs ... Years P-2. The Wonderful World of Combinatorics ... Years 3-5....
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Huffington Post
03 Aug 2015
Thoughts of mathematics are never far from my mind - a hazard of the profession as mathematician, perhaps ... What can I say? I can't help myself ... Or the hints of "combinatorics" in the resolution of the question of "how many croissants did we get?" ... Number theory, combinatorics, geometry, topology, dynamics, all going on in front of me ... Combinatorics underlies much of real-time scheduling and operations research and database management ... ....
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Public Technologies
22 Jul 2015
(Source. West Virginia University) ... The symposium takes place Thursday, July 23 at the Erickson Alumni Center from 11.30 a.m ... NanoSAFE Research Experiences for Undergraduates (REU); STEM Summer Undergraduate Research Experiences (SURE); WVU Honors-administered SURE ; the Center for Neuroscience Summer Undergraduate Research Internships (SURI); the Brazil Scientific Mobility Program and the Graph Theory and Combinatorics Math REU ... -WVU-....
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J.P. Morgan named American Statistical Association Fellow (Virginia Tech - Virginia Polytechnic Institute and State University)
Public Technologies24 Jun 2015
(Source. Virginia Tech - Virginia Polytechnic Institute and State University). BLACKSBURG, Va., June 24, 2015 - Virginia Tech's J.P. Morgan was named a Fellow of the American Statistical Association (ASA), the nation's preeminent professional statistical society ... His research interests focus primarily on experimental design, combinatorics, and discrete optimization ... Morganstein, ASA president, said ... Morgan will be honored at an Aug ... (noodl....
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Laurier researchers receive over $1.7 million from Natural Sciences and Engineering Research Council (Wilfrid Laurier University)
Public Technologies22 Jun 2015
(Source. Wilfrid Laurier University). WATERLOO - The Natural Sciences and Engineering Research Council (NSERC) awarded 17 Wilfrid Laurier University researchers a total of $1.73 million dollars in funding. Two Laurier students received $175,000 in doctoral scholarships ... A wide range of Laurier research projects received funding, including ... research ... algebraic combinatorics of symmetric functions ($55,000) Ian Hamilton, chemistry ... (noodl....
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Huffington Post
16 Jun 2015
It's very common to talk about raising student achievement. After all, this is where we see gaps. Doing poorly on an assignment or failing a test denotes "poor achievement" and stands out to teachers, parents and society. We naturally want to fix student achievement. But here's the twist ... She worked and talked through the problem for more than 20 minutes, ultimately solving a high school-level combinatorics concept about permutations....
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noodls
12 Jun 2015
(Source. Princeton University) Board approves 17 appointments to Princeton faculty. Posted June 12, 2015; 02.00 p.m. by Ushma Patel, Office of Communications. The Princeton University Board of Trustees has approved the appointments of 17 faculty members, including six full professors and 11 assistant professors. Professors ... at the University of Cambridge and her Ph.D. at Columbia ... Marcus studies algorithms, combinatorics and optimization....
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Yahoo Daily News
11 Jun 2015
Conferred with the Padma Shri award this year, Roy's term as the head of the ISI was set to expire in July ... "There is justified and a reasonable apprehension that the present director Dr B ... in combinatorics and optimization from the University of Waterloo, Roy was appointed as the ISI director in August 2010 ... ....
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Huffington Post
11 Jun 2015
Biologist�Dr. E.O.Wilson spent many countless childhood hours watching ants. His unremitting interest eventually led to his ground-breaking research into sociobiology and biodiversity, becoming renown as the world's leading expert on myrmecology. Dr ... He's received many major mathematical accolades in areas of combinatorics, harmonic analysis, matrix theory, and other areas. Dr ... This motivation was given a name by Dr ... Myths & Realities ... Dr....
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noodls
11 Jun 2015
representation theory of Lie algebras, algebraic groups and quantum groups, algebraic combinatorics, modular representation theory of finite groups, and invariant theory ... the Journal of Algebraic Combinatorics and AMS' Representation Theory, being the dissertation advisor for five doctoral students, and winning multiple outstanding teaching awards....
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The Guardian
26 May 2015
Guzzlers, how did you get on?. Let me first restate the problem. Albert, Bernard and Cheryl became friends with Denise, and they wanted to know when her birthday is. Denise gave them a list of 20 possible dates. Denise then told Albert, Bernard and Cheryl separately the month, the day and the year of her birthday respectively. Albert ... Bernard ... Cheryl ... Albert. Now I know when Denise’s birthday is. Bernard ... In the symbology of combinatorics ... ....
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