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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 Non-increasing 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 ⭐
⌨️ This course is created in collaboration with University of California SAN DIEGO

combinatorics and probability practice problems,
combinatorics and probability
combinatorics and discrete probability