#### 组合工具

系统介绍数学中最有趣的数学科目之一——组合学。虽然组合学在中学标准课程中不进行教授，但却经常出现在数学竞赛的难题中。该课程难度范围覆盖MATHCOUNTS、AMC 8 和 AMC 10、AMC 12、AIME。

想了解我们独具一格的教学方法？visit our home page first，来看看我们是如何打破线上数学课的固定框架的吧！

#### 点击开启麦克风

### 适合参加课程的学生

你是否正在上从6年级数学到微积分学前课的数学课程，并且已经了解综合入门课程中的内容？如果是的话，你已经具备了足够的能力来加入这门课。这门课将带你高效学习如何理解学校课本里并不教授，但却极具挑战性的组合学概念。

分级测试：

### 课程结构

每个模块的课程均包含 20节LIVE课程，分为16节【日挑战】课程和4节【周挑战测验】复习课程（学生需要在复习课前进入配套的Pre-Recorded课程完成周挑战测验）。每节课程都将由 2 位 Stars 共同授课。课程整体包含超过150个数学问题。

每节课中，学生都会学习解决全新的数学问题，老师们将通过启发式的教学引导和鼓励学生开创自己的解题方法。我们的LIVE课堂赞赏学生们提出自己的想法，鼓励每个学生和全班同学一起集思广益，构建各种不同的解题方法，更积极地参与数学发现。

【周挑战测验】包含20道数学问题，其中10题在正常难度范围内，而另外10题则非常具有挑战性。

### 回顾

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

### 课程大纲

**第1课**

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

**第2课**

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

**第3课**

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

**第4课**

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

**第5课**

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

**第6课**

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

**第7课**

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

**第8课**

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

**第9课**

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

**第10课**

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

**第11课**

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

**第12课**

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

**第13课**

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

**第14课**

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

**第15课**

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

**第16课**

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