#### LIVE 135: Combinatorics

수ㆍ금 12:00am–1:30 from 6/19(미국 동부 시간)

이 LIVE 강좌에는 라이브 동영상 강의 30시간과 함께(시간당 20) 이 강좌를 통해 1년간 이용할 수 있는 Loh 교수의 녹화 동영상이 포함되어 있습니다.

We’ve carefully designed our courses to maximize engagement. Each of our LIVE classrooms has at least one staff member for every 13 students. The typical class size is around 30 students.

* The official meeting time for this course is 8:00pm in New York.

진단 시험:

### 캘린더

강조 표시된 날짜에 수업이 열립니다. 날짜를 클릭해 그날의 주제를 살펴보세요.

컬러 키:

새로운 주제 수업

숙제로 주어진 시험 토론

### 교육과정

20개의 강좌 모임은 수업 16개(아래 1일차부터 16일차로 표시됨) 및 숙제 시험 토론 4개로 나뉩니다. 수업 4강 이후 각 시험 토론 모임이 열립니다.

**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