#### 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.

저희의 혁신적인 교수법을 아직 확인하지 못하셨다면, visit our home page first하여 온라인 수학 교육의 모습에 대한 **고정관념을 깨보세요**.

#### 클릭하여 음소거

### 이 강좌는 누구에게 적합한가요?

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.

진단 시험:

### 강좌 구조

이 강좌에는 수업 16개와 이 강좌의 4가지 1시간짜리 연습 시험(숙제)을 논의하는 시험 검토 세션 4개까지, 1시간짜리 LIVE 동영상 세션 20개가 포함되어 있습니다. 각각의 수업은 Stars 2명이 함께 가르칩니다. 총 150개 이상의 문제가 있습니다.

여러분에게 익숙하지 않을 여러 문제를 각 수업에서 다루므로, 나만의 해결 방법을 만들어내도록 영감을 줍니다. LIVE 동영상 수업 형식을 통해 아이디어를 제안할 수 있으며, 수업에서 자체적으로 나온 아이디어를 이용해 대안적 해결 방법을 다양하게 구성할 수 있도록 함께 브레인스토밍합니다. 이러한 활동은 수학을 깨칠 수 있게 여러분을 적극적으로 참여시킵니다.

각 연습 시험에는 20문제가 있고, 절반은 평균 난이도이며 절반은 아주 어렵습니다.

### 리뷰

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

### 교육과정

**1일차**

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

**2일차**

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

**3일차**

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

**4일차**

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

**5일차**

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

**6일차**

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

**7일차**

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

**8일차**

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

**9일차**

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

**10일차**

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

**11일차**

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

**12일차**

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

**13일차**

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

**14일차**

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

**15일차**

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

**16일차**

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

### 등록

이 강좌에는 LIVE 및 자기 진도 맞춤형 버전이 있습니다. LIVE 형식을 강력 추천합니다. 학생들이 도전을 사랑하게 되도록 세심하게 설계되어 있기 때문입니다. 이런 방식은 지금껏 없었습니다. 모든 시간은 미국 동부 표준 시간대를 기준으로 하며, 11월 6일 일요일부터 미국 일광 절약 시간 조정에 따릅니다.

일정에 맞출 수 없거나 이미 특별히 동기가 강한 학생들을 위해, 자기 진도 맞춤형 버전에는 이 주제에 대한 Loh 교수의 가르침이 모두 포함되어 있습니다. 원하는 형식을 선택해 등록하세요.