LIVE 30: Combinatorics
Tu/W/Th/F/M 9:00–10:30pm from 7/12 (Your Time*)
This LIVE course contains a total of 30 hours of live video instruction (~$17/hour), plus 1 year of access to Prof. Loh's recorded videos from this course.
We’ve carefully designed our courses to maximize engagement. Each of our LIVE classrooms has a 1:13 staff to student ratio, with a maximum of 40 students.
* The official meeting time for this course is 5:00pm in New York.
Diagnostic Tests:
Calendar
Classes will meet on the highlighted dates. Click on any date to view the topics for that day.
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Lesson on new topic
Discuss exam assigned for homework
Syllabus
The 20 course meetings are split in 16 lessons (called Day 1 through Day 16 below), and 4 homework exam discussions. Each exam discussion meeting happens after 4 lessons.
Day 1
Permutations; counting with restrictions; counting with symmetry; tree diagram for representing outcomes; casework; counting pairs of objects; correction for overcounting
Day 2
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
Day 3
Variation on Venn diagram; union and intersection of sets; set notation; inclusion-exclusion principle; prime factors; application of counting techniques to Number Theory; divisibility
Day 4
Counting on a grid; casework; patterns in counting; rotation and reflection; rotational symmetry and reflective symmetry; factorials; permutations and combinations; correcting for overcounting
Day 5
Permutations with repeated elements; multiplication principle; factorials; correction for overcounting; casework; binomial coefficients; "choose" notation; rotational and reflective symmetry
Day 6
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
Day 7
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
Day 8
Casework; allocation-of-resource problems and arrangements; complementary counting; permutations with repeated elements; application of binomial coefficients
Day 9
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
Day 10
Tiling problems; recursive sequences; permutations with repeated elements; aₙ notation for elements of a sequence; binomial coefficients and choose notation; case analysis; Fibonacci sequences
Day 11
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
Day 12
Graph theory basics; coloring problems; node, vertex, and graph; case analysis; symmetry; pigeonhole principle; four-color theorem; complementary counting; permutations; tree diagrams
Day 13
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
Day 14
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
Day 15
Application of counting techniques to word problems; shortest path problems; representing states using diagrams; breadth-first search technique
Day 16
Polyhedra vertices, edges and faces; Euler's polyhedral formula and motivation for; correction for overcounting; Platonic solids; stellated dodecahedron
Co-Stars
JESSE BRODTMAN
Leader of state-wide math competition program • Placed top 6 in national math competitions • Perfect score on USACO bronze
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JESSIE WANG
Attended MATHCOUNTS State (TX) twice • AIME qualifier since 7th grade • Enjoys singing in school choir, playing with her dog
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