• University of Padova

Combinatorics: Strategies and Methods for Counting

Explore the wonderful world of combinatorics with this course exploring simple and efficient ways to count.

Combinatorics: Strategies and Methods for Counting

See combinatorics made simple and how to avoid errors in counting principles

Combinatorics is an area of mathematics primarily concerned with counting. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry.

On this free online combinatorics course, you’ll discover a simple and efficient method to translate a combinatorial problem into counting the elements of a reference mathematical structure.

You’ll learn basic counting principles, and be able to explain the most frequent errors in their misuse. Ultimately, you’ll discover that combinatorics isn’t as difficult as it seems.

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You can sign up at any time. However, if you’d like to take advantage of feedback from the educators and your mentors, you can do so within the following periods:

June 1 - July 15

December 1 - January 15

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Skip to 0 minutes and 3 seconds Hello, my name’s Albert. Hello, I’m Carlo. We both are mathematicians of the University of Padua, which is a very closer to Venice. Actually, we are both Venetians. Here are some problems that we are going to consider in this MOOC. How many hands of five cards can be dealt with a deck of 52? And how many have exactly three red cards? like that one, for instance. Or very famous birthday problem, which is the probability that in the class of 24 students, the least two of them share the same birthday? and there’s another one. Then tourists lift their passport at the desk. Afterwards, the clerk gives them back randomly. What is the probability that nobody receives is on passport?

Skip to 0 minutes and 56 seconds Did you know that this value is strictly related to the Euler number? True Carlo Another problem is counting the derangements of a sequence. The derangement is a complete new way of ordering the elements of a sequence. Like, for example, that thederangements of the islands of our city, Venice. We will show you how to solve this kind of problems. Actually, they are all related to combinatorics. That is, counting the number of elements of a set. It sounds easy. But very often, mistakes occur in this type of computations. True Alberto. But after years of teaching this topic, we developed a full proof method. Yes. We translate the problem, say, in counting the number of elements of a precise mathematical structure.

Skip to 2 minutes and 6 seconds And then it’s enough to apply a specific principle and it’s done. Simple! Let us start, follow us and learn how to count.

What topics will you cover?

  • Convert counting problems into precise mathematical models
  • Count the basic combinatorial objects like sequences and collections
  • Applications on counting anagrams, derangements of a sequence (hats problem)
  • Count card shuffles and understand astonishing magic tricks
  • Basic probabilities (birthday problem)
  • Algebraic tools that allow to solve subtle combinatorial problems

When would you like to start?

Start straight away and learn at your own pace. If the course hasn’t started yet you’ll see the future date listed below.

  • Available now

What will you achieve?

By the end of the course, you‘ll be able to...

  • Apply the principles of combinatorics to solve the basic combinatorial problems
  • Model some real life counting problems into that of counting precise mathematical structures
  • Identify the mathematical structure which lies besides a combinatorial problem: sequences, collections, sharings, compositions, partitions, derangements.
  • Identify the principle to face a combinatorial problem: bijiection, multiplication, division
  • Calculate the probability of an event when the sample space is composed by equiprobable elementary events
  • Calculate the number of possible outcomes of an aleatory experiment
  • Calculate the number of sequences of prescribed length, with or without repetitions, from a given alphabet
  • Calculate the number of sets of given cardinality
  • Calculate the number of partitions of a given set in a prescribed number of subsets
  • Calculate the number of permutations of a sequence without repetitions
  • Calculate the number of distributions of distinguishable/indistinguishable objects in a given number of numbered boxes (possibly empty)
  • Calculate the number of distributions of distinguishable/indistinguishable objects in a given number of numbered non-empty boxes

Who is the course for?

This course is ideal for anyone interested in mathematical problems, with a basic background in precalculus. It would be useful for anyone wanting to study or work in mathematics, or anyone who wants to develope their critical thinking and problem-solving skills.

Who will you learn with?

I am a professor of algebra at the University of Padova and I teach at the science and engineering faculties. I like very much math and classical music.

I am professor of mathematical analysis at the University of Padova. Among my interests: magic, tango, photography, innovative teaching, technology and e-learning.

Who developed the course?

University of Padova

The University of Padova is one of Europe’s oldest and most prestigious seats of learning; it aims to provide its students with both professional training and a solid cultural background.

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