IB Maths Combinatorics & Counting Principles: What You Actually Need to Know

You need factorial notation, permutations (nPr), combinations (nCr), and the multiplication and addition principles for arrangements with restrictions — repeated items, fixed positions, grouping. That's the SL core. HL uses these same tools inside the binomial theorem and inside binomial/hypergeometric probability distributions.

A permutation counts arrangements where order matters; a combination counts selections where it doesn't. Choosing a captain and vice-captain from 8 players is a permutation (8P2 = 56) because swapping the two names gives a different outcome. Picking any 2 players for a doubles pair is a combination (8C2 = 28).

Yes — the binomial theorem uses nCr to generate expansion coefficients, so it's really combinatorics applied to algebra. At SL you expand $(a+b{)}^n$ for positive integer n. At HL only, the theorem extends to negative and fractional exponents, examined under the analysis and approaches guide's algebra topic.

It's conceptually short but easy to get wrong under pressure — the maths itself is simple, the reasoning about restrictions isn't. In fifteen years of marking mocks, the same three mistakes appear every session: confusing nCr with nPr, forgetting a restriction, and double-counting identical items.

Combinatorics is examined at both SL and HL, in both Maths AA and Maths AI, under Topic 1: Number and Algebra. The core content — nCr, nPr, counting principles — is identical at both levels; HL simply extends it further into the binomial theorem and probability distributions.

Typical questions: seating/arrangement problems, committee or team selection, password or number-plate counting, and binomial expansion coefficients. Most appear as short Paper 1 (no-calculator) questions worth 4-6 marks, though combinatorics regularly resurfaces inside longer Paper 2 probability questions on binomial or hypergeometric distributions.

Ask one question first: if I swap two chosen items, is the result different? If yes, order matters — use nPr. If the outcome is the same either way, use nCr. Always check second whether repetition is allowed, since that changes the formula entirely.

Draw the slots before you calculate anything. Fix whatever's restricted first — the item that must go first, the pair that must sit together — then count the remaining freedom and multiply. Most errors happen because students calculate before identifying the constraint.

Drill restriction-type problems specifically, not just plain nCr/nPr calculations — that's where marks are actually lost. Work through past-paper Section A questions on counting until identifying order-versus-no-order becomes automatic, then move to mixed probability questions that combine combinatorics with binomial distributions.

Combinatorics appears in both courses. Maths AI leans on the GDC's nCr/nPr functions and applies counting to real-world modelling contexts. Maths AA expects more algebraic derivation, including proving binomial coefficient properties, since AA is the more theory-driven of the two current DP maths courses.

Yes. The core counting skills — nCr, nPr, the multiplication principle — are assessed identically at SL and HL. HL simply layers on more: the generalised binomial theorem and its use inside more complex probability distributions, so HL students see it resurface more often across the syllabus.

Combinatorics itself is a small slice of Topic 1 — but skipping it doesn't just cost one question. According to the IB, the current Mathematics: analysis and approaches guide (first assessed 2021) embeds counting principles directly inside the probability and statistics topic, so weak combinatorics quietly costs marks on multiple papers, not one.