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22 LIVE: Combinatorics

Ma/J/D 8:00–9:30pm de 7/5 (hora del Este de EE. UU.)

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El curso de LIVE tiene un total de 30 horas de formación de video en vivo (~17/hora), más 1 año de acceso a los videos pregrabados del curso del Prof. Loh.

Calendario

Las clases se darán en las fechas resaltadas. Haz clic en cualquier fecha para ver los temas del día.

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Lección de un tema nuevo

Analiza el examen asignado como tarea

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Plan de estudios

Los 20 encuentros del curso se dividen en 16 lecciones (llamadas Día 1 hasta Día 16) y 4 horas para analizar los exámenes de tarea. Cada encuentro para analizar un examen es luego de 4 lecciones.

Día 1

Permutations; counting with restrictions; counting with symmetry; tree diagram for representing outcomes; casework; counting pairs of objects; correction for overcounting

Día 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

Día 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

Día 4

Counting on a grid; casework; patterns in counting; rotation and reflection; rotational symmetry and reflective symmetry; factorials; permutations and combinations; correcting for overcounting

Día 5

Permutations with repeated elements; multiplication principle; factorials; correction for overcounting; casework; binomial coefficients; "choose" notation; rotational and reflective symmetry

Día 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

Día 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

Día 8

Casework; allocation-of-resource problems and arrangements; complementary counting; permutations with repeated elements; application of binomial coefficients

Día 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

Día 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

Día 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

Día 12

Graph theory basics; coloring problems; node, vertex, and graph; case analysis; symmetry; pigeonhole principle; four-color theorem; complementary counting; permutations; tree diagrams

Día 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

Día 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

Día 15

Application of counting techniques to word problems; shortest path problems; representing states using diagrams; breadth-first search technique

Día 16

Polyhedra vertices, edges and faces; Euler's polyhedral formula and motivation for; correction for overcounting; Platonic solids; stellated dodecahedron

Co-Estrellas

Foto de perfil de Alon Ragoler
ALON RAGOLER

National MATHCOUNTS competitor for Washington • Perfect score on AMC 8 • Enjoys web development, playing instruments (piano + snare drum)

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Foto de perfil de Angela Guo
ANGELA GUO

Placed 2nd in Illinois State MATHCOUNTS Countdown Round • MPFG invitee • USESO camper, enjoys kayaking, playing viola, making videos

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