Introduction
Combinatorics is the branch of Discrete Mathematics that answers one deceptively simple question: "In how many ways can something happen?" Instead of doing the counting one by one (which is impossible when the numbers are huge), combinatorics gives you formulas and principles to count without listing.
Why it matters for the exam: counting problems appear in almost every competitive test — BCS (preliminary math & mental ability), bank officer (quantitative aptitude), and IT-officer posts (discrete math, probability foundations). Questions on arrangements, selections, committees, handshakes, number-plate codes, seating, and probability all rest on the ideas in this lesson. Master these formulas and you can solve a whole family of questions in seconds.
Learning objectives
- Apply the Rule of Sum and the Rule of Product correctly and know when to use each.
- Compute factorials, permutations (ⁿPᵣ) and combinations (ⁿCᵣ) and clearly distinguish "order matters" from "order does not matter".
- Handle repetition, circular arrangements, and permutations with identical objects.
- Use the binomial theorem and Pascal's triangle to expand and to count.
- Apply the Pigeonhole Principle and the Inclusion–Exclusion Principle to solve tricky counting problems.
- Recognise the standard exam scenarios (handshakes, diagonals, committees, code words) and pick the right tool instantly.