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！
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这门课适合我吗？
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。
选择课程前，来挑战一下能力水平测试吧！
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。在该模块的学习中，我们将讨论超过150个数学问题。
【日挑战课程】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。
学生评价
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 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 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 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 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 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 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 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 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 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
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
报名课程
每位报名LIVE在线直播课的学生都将同时享有对应的Pre-Recorded自学录播课程一年使用权，无需额外购买。LIVE直播课经过罗教授和教学团队精心设计的，通过前所未有的引人入胜的教学方式、极为丰富的课堂交互体验，让学生在趣味学习中爱上挑战和思考。 以下所有直播课显示时间均为您的当地时间（美国将于2023年3月12日星期日进入夏令时，请其他国家和地区的学生注意课程时间调整）。
如果你有强大的自驱力可以进行自主学习，或者想比LIVE课程安排更快地完成一个模块的学习，也欢迎你选择我们的自学录播课程。录播课包含了所有罗教授关于该模块的教学视频（没有直播课堂），在过去3年中也有数千名来自世界各地的同学参与学习。