Topics:

Basic counting rules (product rule, sum rule, permutations, combinations, Pascal's triangle, occupancy problems).

Introductory graph theory (fundamental concepts, connectedness, graph coloring, trees, Cayley's theorem on the number of trees).

Generating functions (definition, operations on generating functions, applications to counting, binomial theorem,
exponential generating functions).

Recurrences (Fibonacci numbers, derangements, recurrences involving more than one sequence, the method of generating functions).

Principle of inclusion and exclusion (the principle and applications, occupancy problems with distinguishable balls and
cells, derangements, the number of objects having exactly $m$ properties).

Pigeonhole principle and Ramsey theory (Ramsey's theorem, bounds on Ramsey numbers, applications).

Symmetric combinatorial structures (block designs (definition, latin  squares, finite  projective planes).

Remark.If there will me more then 20 students interested in this course, course  Combinatorics 1 (B) will introduced  as well with a similar syllabus but different (faster) pace. In COB we assume that students are basically familiar with most of the basic counting rules, factorials, combinations,, etc and will reach to the end of the syllabus. In COA the last topic - symmetric structures - may be omitted.