组合工具图标

LIVE 30: 组合工具

二、三、四、五、一17:00–18:30从7月12日 (美国东部时间)

报名课程 $519课程详细介绍

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。

课程日历

上课日期由彩色图例标识。点击任意上课日期了解当天课程内容。

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

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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

启明星助教老师

Jesse Brodtman 的头像
JESSE BRODTMAN

州级别数学竞赛项目领队 • 全国级别数学竞赛前六名 • USACO(美国计算机奥林匹克)铜牌满分获得者 •

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Jessie Wang 的头像
JESSIE WANG

两次入围德克萨斯州MATHCOUNTS州级赛 • 七年级即获得AIME参赛资格 • 参加学校合唱团,喜欢唱歌,也喜欢和她的小狗一起玩

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