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.

This course is part of the Introduction to Discrete Mathematics for Computer Science Specialization

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## About this Course

### Learner Career Outcomes

## 33%

## 50%

## Skills you will gain

### Learner Career Outcomes

## 33%

## 50%

### Offered by

#### University of California San Diego

UC San Diego is an academic powerhouse and economic engine, recognized as one of the top 10 public universities by U.S. News and World Report. Innovation is central to who we are and what we do. Here, students learn that knowledge isn't just acquired in the classroom—life is their laboratory.

#### National Research University Higher School of Economics

National Research University - Higher School of Economics (HSE) is one of the top research universities in Russia. Established in 1992 to promote new research and teaching in economics and related disciplines, it now offers programs at all levels of university education across an extraordinary range of fields of study including business, sociology, cultural studies, philosophy, political science, international relations, law, Asian studies, media and communicamathematics, engineering, and more.

## 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.

**3 hours to complete**

**12 videos**

**4 readings**

**8 practice exercises**

**4 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!

**4 hours to complete**

**8 videos**

**4 readings**

**6 practice exercises**

**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.

**3 hours to complete**

**8 videos**

**3 readings**

**8 practice exercises**

**5 hours to complete**

## Probability

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.

**5 hours to complete**

**17 videos**

**4 readings**

**10 practice exercises**

## Reviews

### TOP REVIEWS FROM COMBINATORICS AND PROBABILITY

Great course. The final Project unclear had instructions on how to provide input. I spent a lot of time trying to troubleshoot it even though I already have a correct solution

This course provided me with new ways to confront the problems of combinatorics. I am very grateful to the faculty for their content and coursera for giving me financial aid.

Great course, lots of good info, not too long. Some of the coding assignments and quizzes are challenging, but the staff respond very quickly to questions in the forums.

Had loads of fun during most part of the course. Frequent quizzes keep the learner on toes. Thoroughly enjoyed the final programming quiz to implement a dice game.

## About the Introduction to Discrete Mathematics for Computer Science Specialization

## Frequently Asked Questions

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