About this Course
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Beginner Level


Subtitles: English, Greek, Chinese (Simplified)

Skills you will gain

Random VariableProbability InterpretationsProbabilityCombinatorics

100% online

Start instantly and learn at your own schedule.

Flexible deadlines

Reset deadlines in accordance to your schedule.

Beginner Level


Subtitles: English, Greek, Chinese (Simplified)

Syllabus - What you will learn from this course

3 hours to complete

Basic Counting

Suppose we need to count certain objects. Can we do anything better than just list all the objects? Do we need to create a list all phone numbers to check whether there are enough phone numbers for everyone? Is there a way to tell whether 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 module we will give an introduction to this field that will help us to answer basic versions of the above questions.

12 videos (Total 54 min), 4 readings, 9 quizzes
12 videos
Convenient Language: Sets4m
Generalized Rule of Sum3m
Number of Paths4m
Rule of Product3m
Back to Recursive Counting3m
Number of Tuples5m
Licence Plates3m
Tuples with Restrictions5m
4 readings
Listing All Permutations5m
8 practice exercises
Rule of Sum in Programming4m
Numbers Divisible by 2 or 38m
Operations with Sets10m
Generalized Rule of Sum18m
Rule of Product in Programming10m
Applications of the Rule of Product12m
Counting with Restrictions20m
3 hours to complete

Binomial Coefficients

In how many ways one can select a team of five students out of ten students? What is the number of non-negative integers with at five digits whose digits are decreasing? In how many ways one can get from the bottom left cell to the top right cell of a 5x5 grid, each time going either up or to the right? And why all these three numbers are equal? We'll figure this out in this module!

8 videos (Total 76 min), 4 readings, 6 quizzes
8 videos
Pascal's Triangle9m
Row Sums10m
Binomial Theorem12m
Practice Counting13m
4 readings
Generating Combinatorial Objects: Code10m
6 practice exercises
Number of Segments and Diagonals20m
Forming Sport Teams15m
Number of Iterations of Nested For Loops4m
Sum of the First Six Rows of Pascal's Triangle2m
Expanding (3a-2b)^k20m
Practice Counting10m
3 hours to complete

Advanced Counting

We have already considered most of the most standard settings in Combinatorics, that allow us to address many counting problems. However, successful application of this knowledge on practice requires considerable experience in this kind of problems. In this module we will address the final standard setting in our course, combinations with repetitions, and then we will gain some experience by discussing various problems in Combinatorics.

8 videos (Total 36 min), 3 readings, 8 quizzes
8 videos
Distributing Assignments Among People3m
Distributing Candies Among Kids3m
Numbers with Fixed Sum of Digits4m
Numbers with Non-increasing Digits2m
Splitting into Working Groups4m
3 readings
8 practice exercises
Combinations with Repetitions10m
Distributing Assignments Among People10m
Distributing Candies Among Kids15m
Numbers with Fixed Sum of Digits15m
Numbers with Non-increasing Digits7m
Splitting into Working Groups10m
Problems in Combinatorics45m
5 hours to complete


The word "probability" is used quite often in the everyday life. However, not always we can speak about probability as some number: for that a mathematical model is needed. What is this mathematical model (probability space)? How to compute probabilities (if the model is given)? How to judge whether the model is adequate? What is conditional probability and Bayes' theorem? How our plausible reasoning can be interpreted in terms of Bayes' theorem? In this module we cover these questions using some simple examples of probability spaces and real life sutiations.

17 videos (Total 126 min), 4 readings, 11 quizzes
17 videos
Rolling Dice7m
More Probability Spaces10m
Not Equiprobable Outcomes4m
More About Finite Spaces6m
Mathematics for Prisoners7m
Not All Questions Make Sense10m
What Is Conditional Probability?7m
How Reliable Is The Test?8m
Bayes' Theorem8m
Conditional Probability: A Paradox7m
Past and Future8m
Monty Hall Paradox8m
`Our Position'6m
4 readings
10 practice exercises
Concentration for Galton Board10m
Computing Probabilities for Two Dice12m
Computing Probabilities: More Examples12m
Fair Decisions and Imperfect Coins20m
Inclusion-Exclusion Formula10m
Computing Conditional Probabilities16m
Prisoner, King and Conditional Probabilities10m
More Conditional Probabilities8m
More About Independence20m
Monty Hall Gone Crazy20m
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Top reviews from Combinatorics and Probability

By ZBOct 13th 2018

I really enjoyed taking this course. The teaching was pretty good and some of the quiz questions will challenge you if you haven't done Combinatorics before.

By CZSep 11th 2018

The final project is hard for me cuz I don't have Python experience. and the logic is a little bit complicated. That's not for absolutely beginners!



Alexander S. Kulikov

Visiting Professor
Department of Computer Science and Engineering

Vladimir Podolskii

Associate Professor
Computer Science Department

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