Introduction
This lesson gathers the "second-tier" discrete mathematics topics that appear again and again in Bangladeshi government job examinations (BCS, Bangladesh Bank / combined bank IT-officer, and various Assistant Programmer / Computer Operator posts) but are often skipped in a first pass. Each topic here is a self-contained tool: a formula, a method, or a puzzle-solving trick that turns a "hard-looking" MCQ or short question into a two-minute win. We cover advanced counting (stars and bars, positional permutations, pigeonhole), logic (categorical syllogisms, knights and knaves), set theory (membership-table proofs, set reconstruction, inclusion–exclusion bounds), recurrence relations, and graph theory (Euler's planar formula, chromatic number, edge coloring, isomorphism).
Why it matters for the exam: these are exactly the items that separate a 6/10 from a 10/10 in the mathematics/reasoning section, because they are procedural. Once you know the method, the answer is forced — there is little "insight" left to gamble on.
Learning objectives
- Distribute identical objects into distinct boxes using stars and bars, including the "at least one each" variant.
- Test a categorical syllogism for validity using the Venn-diagram method.
- Solve knights and knaves truth-teller puzzles by case analysis.
- Prove a set identity (De Morgan on three sets) with a membership table.
- Reconstruct sets and bound cardinalities using union, intersection, difference and inclusion–exclusion.
- Set up and solve recurrence relations (Fibonacci, bit strings with no two consecutive 0s, and a general linear recurrence).
- Count permutations with positional constraints (vowels in odd positions).
- Apply the pigeonhole principle to guarantee a monotonic subsequence.
- Use Euler's formula V − E + F = 2 and its planarity corollaries.
- Find the chromatic number and edge-chromatic number of small graphs.
- State and apply the necessary conditions for graph isomorphism.