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How To Say Combinatorics
นอกจากการดูบทความนี้แล้ว คุณยังสามารถดูข้อมูลที่เป็นประโยชน์อื่นๆ อีกมากมายที่เราให้ไว้ที่นี่: ดูความรู้เพิ่มเติมที่นี่
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Combinatorial Geometry
Combinatorics
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 mathematical 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.
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Combinatorics: Factorials Explained
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This is going to be a short optional lecture explaining factorials. The notation ‘n factorial’ is used to
express the product of the natural numbers from 1 to n. This means that n factorial equals 1, times
2, times 3, all the way up to n. For instance, 3 factorial is equal to 6, since: 1, times 2, times 3, equals 6 Simple enough, right? For the remainder of the lecture, we are going to explain some important properties of factorial mathematics. Before we get into the more complicated concepts,
you should know that there is one odd characteristic: negative numbers don’t have a factorial,
and zero factorial is equal to 1 by definition.
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combinatorics Statistics Factorial
Combinatorics and Probability (Complete Course) | Discrete Mathematics for Computer Science
TIME STAMP
BASIC COUNTING
0:00:00 Why counting
0:02:58 Rule of Sum
0:06:33 How Not to Use the Rule of Sum
0:10:06 Convenient Language Sets
0:15:01 Generalized Rule of Sum
0:18:46 Numbers of Paths
0:23:39 Rule of Product
0:26:44 Back to Recursive Counting
0:30:32 Number of Tuples
0:35:48 Licence Plates
0:39:20 Tuples with Restrictions
0:44:24 Permutations
BINOMIAL COEFFICIENTS
0:53:53 Previously on Combinatorics
0:59:44 Number of Games in a Tournament
1:10:39 Combinations
1:19:11 Pascal’s Traingle
1:29:08 Symmetries
1:33:16 Row Sums
1:44:13 Binomial Theorem
1:57:06 Practice Counting
ADVANCED COUNTING
2:10:24 Review
2:14:11 Salad
2:19:21 Combinations with Repetitions
2:27:17 Distributing Assignments Among People
2:30:55 Distributing Candies Among Kids
2:34:35 Numbers with fixed Sum of Digits
2:39:26 Numbers with Nonincreasing Digits
2:42:01 Splitting into Working Groups
PROBABILITY
2:46:13 The Paradox of Probability Theory
2:50:16 Galton Board
2:56:43 Natural Sciences and Mathematics
3:02:51 Rolling Dice
3:10:24 More Probability Spaces
3:20:48 Not Equiprobable Outcomes
3:25:35 More About Finite Spaces
3:31:59 Mathematics for Prisoners
3:39:37 Not All Questions Make Sense
3:49:40 What is Conditional Probability
3:57:02 How Reliable Is The Test
4:05:28 Bayes’Theorem
4:14:06 Conditional Probability A Paradox
4:21:30 past and Future
4:29:32 Independence
4:37:35 Monty Hall Paradox
4:46:04 our Position
RANDOM VARIABLES
4:52:28 Random Variables
4:54:30 Average
4:59:41 Expectation
5:09:10 Linearity of Expectation
5:16:51 Birthday Problem
5:27:14 Expectation is Not All
5:32:08 From Expectation to Probability
5:34:55 Markov’s Inequality
5:42:07 Application to Algorithms
PROJECT: DICE GAMES
5:46:51 Dice Game
5:50:13 Playing the GAme
5:58:35 project Description
Counting is one of the basic mathematically related tasks we encounter on a day to day basis. The main question here is the following. If we need to count something, can we do anything better than just counting all objects one by one? Do we need to create a list of all phone numbers to ensure that there are enough phone numbers for everyone? Is there a way to tell that our algorithm will run in a reasonable time before implementing and actually running it? All these questions are addressed by a mathematical field called Combinatorics.
In this course we discuss most standard combinatorial settings that can help to answer questions of this type. We will especially concentrate on developing the ability to distinguish these settings in real life and algorithmic problems. This will help the learner to actually implement new knowledge. Apart from that we will discuss recursive technique for counting that is important for algorithmic implementations.
One of the main `consumers’ of Combinatorics is Probability Theory. This area is connected with numerous sides of life, on one hand being an important concept in everyday life and on the other hand being an indispensable tool in such modern and important fields as Statistics and Machine Learning. In this course we will concentrate on providing the working knowledge of basics of probability and a good intuition in this area. The practice shows that such an intuition is not easy to develop.
In the end of the course we will create a program that successfully plays a tricky and very counterintuitive dice game.
⭐ Important Notes ⭐
⌨️ The creator of this course is University of California SAN DIEGO
⌨️ For earning certificate, enroll for this course here: https://www.coursera.org/specializations
combinatorics and probability practice problems,
combinatorics and probability
combinatorics and discrete probability
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