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Combinatorics (University Level) : Counting Principles
Rating: 4.8 out of 5(120 ratings)
1,668 students

Combinatorics (University Level) : Counting Principles

Counting for Data Science, GRE, DAT, GMAT, probability, and combinatorics!
Created byMath and Science Academy
Last updated 7/2023
English
English [Auto],Korean [Auto],

What you'll learn

  • You will be able to handle counting problems necessary for tests such as the GRE, GMAT, DAT, etc.
  • You will have a firm foundation for pursuing more combinatorics.
  • You will be comfortable with permutations, combinations, binomial coefficients, ordering with repetition, choosing with repetition, PIE, etc.
  • You will be comfortable with the combinatorics needed for beginning a data science career.

Explore related topics

Course content

6 sections • 29 lectures • 5h 42m total length

  • Introduction1:17

    This short, self-contained course gives you the basics on counting necessary to be successful in probability, stats, genetics, GMAT, GRE, etc.  The multiplication principle, additional rule, permutations, combinations, counting strategies, binomial coefficients, multinomial coefficients, inclusion & exclusion, and derangements are covered.  In a follow up course, we will cover more involved topics such as generating functions and Polya's counting theory.  Abbreviations for applicability to different tests: DAT, GRE, GMAT, MSG (Math Subject GRE), OAT (O), ISSE (I).

  • The Multiplication Rule (DAT, GRE, GMAT, O, MSG, I)8:11

    Apply the counting principle, i.e., the multiplication rule, to multiply options at each stage, as shown by choosing hats, shirts, and shoes to form outfits.

  • The Multiplication Rule
  • Distinguishable vs Indistinguishable (Optional, MSG)10:22

    Explain distinguishable versus indistinguishable objects with examples, showing how the counting principle depends on whether items can be told apart or look identical.

  • Permutations (DAT, GRE, GMAT, O, MSG, I)9:23

    Explore permutations with no replacement and where order matters, using the counting principle to assign president, vice president, and treasurer from ten distinguishable people, and derive n!/(n-k)!.

  • Permutation Problems (DAT, GMAT, GRE, O, MSG, I)8:27
  • Permutations
  • Multinomial (DAT, GRE, GMAT, O, MSG)8:50

    Learn how to count arrangements with repeats by applying permutations and factorials, dividing by repeated-letter factorials to correct overcount, introducing the multinomial principle.

  • Multinomial
  • Combinations (DAT, GRE, GMAT, O, MSG, I)13:59
  • Combination Problem (Large Lecture Format)2:37

    Here is a quick bonus problem taken from a class lecture John gave.

  • Combinations
  • Bonus: Practice with Probability!5:44

    In this bonus lesson, we will apply our knowledge of combinatorics to probability. For the problem we look at, it is enough to know that the probability can be computed by diving the number of ways of getting "what we want" by the "total number of possibilities" [WANT/TOTAL]. Here, we take for granted that each possibility is equally likely to occur.

  • Bonus: A Faster Way to Do the Round Table Problem (Lecture 9)!9:54
  • Binomial Coefficients (MSG)12:35
  • Bonus Problem: An Application with Nonconsecutive Numbers!12:58

    A bonus lecture showing two interesting ways of picking 3 nonconsecutive numbers from the numbers 1-10!

Requirements

  • You should be comfortable with arithmetic and basic algebra skills.
  • You should be comfortable with US college level work (pacing), e.g., UCI, UCLA, USC, etc.
  • You should have enough mathematical maturity to stick with a "hard" topic. Combinatorics can seem easy at first, but it can easily stump you with "difficult" questions that look so simple.
  • You should have some college level math experience (calculus) OR be just be REALLY dedicated.
  • This course can be a breeze or can be very difficult based on your interests and, most importantly, your dedication.

Description

Learn the multiplication rule, permutations, combinations, n choose r with repetition, multinomial, the principle of inclusion and exclusion, partitions, and derangements.  John will take you through the ideas and techniques you need to get a firm handle on counting concepts and applications.  This course is perfect for people wanting to learn counting strategies for tests such as the GRE, DAT, and GMAT, for anyone interested in Data Science, for anyone studying combinatorics , probability, or statistics, and for those just interested in interesting enumeration problems.   John mentions some extra applications that require some knowledge of numbers such as "e," but these applications can easily be skipped with no loss of continuity.

Combinatorics is a growing field utilized in data science, computer science, statistics, probability, engineering, physics, business management, and everyday life.  This course is a great introduction with some specialized topics.  It is best for someone getting started.  If you are more experienced, this course is not for you unless you want to revisit the core concepts.  Please see the list of topics.

*Although this course covers only the topics listed, the material can be challenging and demand time to fully absorb!

**This is paced as "Beginner" for a US College Course. But, on UDEMY the course rated 5.0 until a student rated the it a 1.0 due to the difficulty. A few other students commented that they loved the course but that it was set at a difficulty level greater than "Beginner" with respect to many UDEMY courses. To be consistent with the platform, the class has been adjusted to an "Intermediate" level course. However, if you are taking a university course in the US on combinatorics, this course would be the equivalent of the first part of a combinatorics class.

Who this course is for:

  • This course starts from scratch and covers counting for probability and combinatorics at an introductory level, BUT at a "US college" pace.
  • Example: Someone studying for the GRE or DAT, studying computer science, starting a combinatorics course, studying statistical mechanics, etc.
  • This course is not for someone with experience in combinatorics (unless you want to review core counting).
  • This course is for anyone with an interest in combinatorics! After all, we all count!

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