LIVE 30: 组合工具
7月12日起，每周二、三、四、五、一下午9:00–10:30 （您所在地当地时间）
该模块直播课总计30小时，课时费约为(~$17/小时)。报名直播课将同时享受该模块对应录播课程（Pre-Recorded Course）一年有效，无限次回看。
经过罗教授和课程团队反复调整，我们选择以不低于1:13的师生比进行授课。老师们将用各种方式鼓励孩子们发言、提问和思考，最大限度地提高孩子们的课堂参与度和积极性。每个班次不超过40名学生。
* 该班次直播课上课标准时间为下午5:00，纽约时间。
选择课程前，来挑战一下能力水平测试吧！
课程日历
上课日由彩色图例标识。点击任意上课日了解当天课程内容
图例
16节日挑战课程，每节课通过不同的数学问题学习新的知识内容
4节周挑战复习课程，集中解决周挑战测试中遇到的问题，学生需要在复习课程前进入Pre-Recorded课程完成周挑战测验
课程大纲
LIVE在线直播课程每个模块包含16次日挑战Session（第1课到第16课）+ 4次周挑战复习课（Weekly Challenge 1到4）共20节课程。每四节日挑战Session结束后将有一次周挑战复习课，学生需要在复习课前进入Pre-Recorded课程完成周挑战测验。
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
