#### Combinatorics

A systematic introduction to one of the most interesting math subjects which is not taught in the standard school curriculum, but which often appears in hard problems on math contests. Covers concepts in MATHCOUNTS, AMC 8, AMC 10, AMC 12, and AIME.

If you haven't yet seen our revolutionary teaching method, visit our home page first, to **shatter your expectations** of what online math looks like.

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### Who Is This Course For?

Are you taking any math class from 6th Grade Math through Pre-Calculus, and already know the content in our Introduction class? If so, then you are already ready to understand this course. It will efficiently teach you the challenging concepts in Combinatorics which are not taught in regular school curriculum.

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### Course Structure

This course contains 20 LIVE 1-hour video sessions, of which 16 are lessons, and 4 are exam review sessions that discuss the course's 4 practice 1-hour exams (homework). Each class is co-taught by 2 of our Stars. In total, there are over 150 problems.

Each lesson contains several problems which are meant to be unfamiliar to you, so that they inspire you to invent your own solutions. In our LIVE video class format, you propose your ideas, and we brainstorm together with you to construct a variety of alternative solutions using the class's own ideas. This actively involves you in discovering mathematics.

Each practice exam contains 20 problems, of which half are of average difficulty, and the other half are very challenging.

### Reviews

This is definitely one of the best math courses available. Prof. Loh explains the topics in a detailed way, and makes it fun.

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is definitely one of the best math courses available. Prof. Loh explains the topics in a detailed way, and makes it fun.

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is definitely one of the best math courses available. Prof. Loh explains the topics in a detailed way, and makes it fun.

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

This is an awesome course.

The video explanations are crystal clear and the challenges are fun. 5/5 would do again.

I think that this course was an excellent tool to “skim the surface” in the world of Combinatorics!

It tells every way to solve a problem and for problems, you get a turn to solve and get hints!

such big brain

### Syllabus

**Day 1**

Permutations; counting with restrictions; counting with symmetry; tree diagram for representing outcomes; casework; counting pairs of objects; correction for overcounting

**Day 2**

Venn diagram; combinations; number of subsets; patterns in counting; sum of consecutive powers of 2; multiplication principle; complementary counting; counting lists of numbers with restrictions; overlapping groups

**Day 3**

Variation on Venn diagram; union and intersection of sets; set notation; inclusion-exclusion principle; prime factors; application of counting techniques to Number Theory; divisibility

**Day 4**

Counting on a grid; casework; patterns in counting; rotation and reflection; rotational symmetry and reflective symmetry; factorials; permutations and combinations; correcting for overcounting

**Day 5**

Permutations with repeated elements; multiplication principle; factorials; correction for overcounting; casework; binomial coefficients; "choose" notation; rotational and reflective symmetry

**Day 6**

Binomial coefficients; Pascal's triangle; symmetry of binomial coefficients; patterns in Pascal's triangle; Pascal's identity; comparing binomial coefficients; hockey stick identity; combinations; casework

**Day 7**

Binomial thm; Pascal's triangle, row sum of and relation to powers of 2; symmetry of binomial coefficients; number of subsets; powers of 11; applications of Binomial thm

**Day 8**

Casework; allocation-of-resource problems and arrangements; complementary counting; permutations with repeated elements; application of binomial coefficients

**Day 9**

Paths on a grid; using diagrams; permutations with repeated elements; factorials; binomial coefficients; complementary counting; reduction of a problem into subproblems; symmetry; difference of squares

**Day 10**

Tiling problems; recursive sequences; permutations with repeated elements; aₙ notation for elements of a sequence; binomial coefficients and choose notation; case analysis; Fibonacci sequences

**Day 11**

Correction for overcounting; patterns in counting; case analysis; counting with restrictions; multiple recursions; applications to recursion and tiling problems; general form of a recursive formula

**Day 12**

Graph theory basics; coloring problems; node, vertex, and graph; case analysis; symmetry; pigeonhole principle; four-color theorem; complementary counting; permutations; tree diagrams

**Day 13**

Counting ordered lists; case analysis; triangular numbers and their relationship to binomial coefficients; hockey stick identity; Pascal's triangle; ways to partition N objects (stars and bars); number of subsets

**Day 14**

Committee-type problems and ways to form pairs; correction for overcounting; permutations; factorials; multiplication principle; double factorial notation; applications of tiling techniques to word problems

**Day 15**

Application of counting techniques to word problems; shortest path problems; representing states using diagrams; breadth-first search technique

**Day 16**

Polyhedra vertices, edges and faces; Euler's polyhedral formula and motivation for; correction for overcounting; Platonic solids; stellated dodecahedron

### Enroll

We have LIVE and self-paced versions of this course. We strongly recommend the LIVE format, as it has been carefully designed to be so engaging that students fall in love with challenge. There has never been anything like it before. All times are in the USA Eastern Time Zone, and will follow the USA Daylight Savings Time adjustment on Sun Nov 6.

For students who cannot match our schedule, or who are already particularly motivated, our self-paced version contains all of Prof. Loh's teaching on this subject. Enroll by choosing the format you prefer: