api: "YAML JSON TOON Database"
version: "1.0.0"
format: "json"
dataset:
  slug: "combinatorics-fundamentals"
  title: "Combinatorics Fundamentals"
  description: "Core counting principles and combinatorial formulas."
  category: "Mathematics"
  category_slug: "mathematics"
  tags: "math,combinatorics,counting,permutations,combinations"
  view_count: 0
data:
  - principle: "Product Rule"
    description: "If task A has m ways and task B has n ways, both can be done in m * n ways."
    formula: "m * n"
  - principle: "Sum Rule"
    description: "If tasks are mutually exclusive with m and n ways, total is m + n."
    formula: "m + n"
  - principle: "Permutations"
    description: "Ordered arrangements of r elements from n items."
    formula: "P(n,r) = n! / (n - r)!"
  - principle: "Combinations"
    description: "Unordered selections of r elements from n items."
    formula: "C(n,r) = n! / (r!(n-r)!)"
  - principle: "Permutations with Repetition"
    description: "Arrangements when some elements are identical."
    formula: "n! / (n1! * n2! * ... * nk!)"
  - principle: "Pigeonhole Principle"
    description: "If n items are placed into m containers and n > m, some container has >= 2 items."
    formula: "ceil(n/m)"
  - principle: "Binomial Theorem"
    description: "Expansion of (a + b)^n."
    formula: "(a+b)^n = Sum C(n,k) * a^(n-k) * b^k"