Combinatorics Decision Framework
Compact guide for selecting the correct counting formula and validating results when working with arrangements and selections.
1. Core Identities
- Permutation: nPr = n!/(n−r)! (ordered)
- Combination: nCr = n!/(r!(n−r)!) (unordered)
- Repetition Comb: C(n+r−1, r)
- Perm w/ Repetition: n^r
- Multinomial: n!/(n₁! n₂! … nₖ!)
- Circular: (n−1)! (fix anchor); if reflections same: (n−1)!/2
- Derangements (approx): !n ≈ n!/e (exact: n! Σ (-1)^k/k!)
2. Classification Flow
- Order matters? → yes: permutation; no: combination
- Repetition allowed? → choose repetition variant
- Identical items? → divide by factorials of repeats
- Circular symmetry? → use (n−1)! adjustment
3. Heuristics
- Use symmetry: nCr = nC(n−r)
- “At least one” = total − none
- Exploding numbers? Recheck if order truly matters
4. Quality Checks
- r>n & no repetition ⇒ 0 ways
- Baselines: nC0=1, nC1=n, nCn=1
- Reduce factorial ratios before expansion
- Compare scale with bounds (e.g., n^r upper bound for order)
5. Common Pitfalls
- Counting rotations separately
- Forgetting identical objects (overcount)
- Misreading wording: arrangement vs selection
- Confusing “with replacement” vs “reorder allowed”
6. FAQ
- Why divide by r!? Removes internal ordering duplicates in combinations.
- Why (n−1)! for circles? Rotations collapse to a single canonical arrangement.
- Which for passwords? Usually permutation with repetition (n^r).
7. Action Tip
Tag each problem explicitly: (order / repetition / symmetry / identical) → translate tag to formula before computing; prevents 80% of miscounts.