Course Materials for Students

Ordinary and strong induction: how to write the proof, how many base cases you need, and the common mistakes.

Rules of sum and product, permutations, combinations, arrangements with repeated objects, complements, the gap method, and combinatorial arguments.

Routes on a road map, identical versus labelled containers, “at least” conditions, stars and bars, and proofs by induction (divisibility, inequalities, sums).

Assignment 1, Q1 to 7: books on shelves, round tables, card hands, strings with a fixed composition, the multinomial theorem, binomial sums, and slack variables.

Assignment 1, Q8 to 14: positive integer solutions, systems of equations, induction for merge sort and the triangle inequality, Fibonacci identities, ascents, and palindromes.

Assignment 2, Q1 to 7: inclusion–exclusion with upper bounds, “exactly m” and “at least m”, glued letters, derangements, generating functions, and compositions of an integer.

Assignment 2, Q8 to 14: generating functions for weighted equations and partitions, exponential generating functions, solving recurrences, and combinatorial proofs.

Graph theory basics (Sections 11.1 to 11.3): directed and undirected graphs, trails, paths and cycles, connectivity, subgraphs and complements, graph isomorphism, and vertex degree.

The Handshaking Theorem, complete and regular graphs, Eulerian trails and Euler’s theorem, a lower bound on the edges of a connected graph, bipartite and planar graphs (K5 and K3,3), and forcing a triangle.

Getting started with draw.io: the interface, the course’s ER notation in draw.io, building and styling a diagram step by step, exporting, common mistakes, and a hands-on exercise.