{
    "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.",
        "summary": "The counting principles used to enumerate possibilities: the sum and product rules, permutations and combinations (with and without repetition), the pigeonhole principle and the binomial theorem, plus inclusion-exclusion, derangements, stars and bars, Catalan and Stirling numbers. Each entry has the principle, a one-line description and the formula.",
        "use_when": [
            "You need the right counting formula for a probability or algorithm-analysis problem.",
            "You are checking whether a count is ordered or unordered, with or without repetition."
        ],
        "caveats": [
            "Formulas assume distinguishable items unless stated; identical items need the multinomial or repetition variants.",
            "Notation varies: C(n, k), nCk and the binomial coefficient all mean the same count."
        ],
        "related": [
            "statistics-basics",
            "set-theory-basics",
            "complexity-classes"
        ],
        "category": "Mathematics",
        "category_slug": "mathematics",
        "tags": "math,combinatorics,counting,permutations,combinations",
        "view_count": 0,
        "entry_count": 16,
        "fields": [
            {
                "section": "data",
                "field": "principle",
                "type": "string",
                "example": "Product Rule"
            },
            {
                "section": "data",
                "field": "description",
                "type": "string",
                "example": "If task A has m ways and task B has n ways, bot…"
            },
            {
                "section": "data",
                "field": "formula",
                "type": "string",
                "example": "m * n"
            }
        ]
    },
    "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"
        },
        {
            "principle": "Inclusion-Exclusion Principle",
            "description": "Counts the union of sets by adding single sets, subtracting pairwise overlaps, adding triple overlaps, and so on.",
            "formula": "|A ∪ B| = |A| + |B| − |A ∩ B|"
        },
        {
            "principle": "Combinations with Repetition",
            "description": "Choose k items from n types where an item can be chosen more than once and order does not matter.",
            "formula": "C(n + k - 1, k)"
        },
        {
            "principle": "Stars and Bars",
            "description": "Counts ways to distribute n identical items into k distinct bins.",
            "formula": "C(n + k - 1, k - 1)"
        },
        {
            "principle": "Multinomial Coefficient",
            "description": "Counts arrangements of n items with k1, k2, ... km identical items of each kind.",
            "formula": "n! / (k1! k2! ... km!)"
        },
        {
            "principle": "Circular Permutations",
            "description": "Arrangements of n distinct objects around a circle, where rotations count as the same arrangement.",
            "formula": "(n - 1)!"
        },
        {
            "principle": "Derangements",
            "description": "Permutations in which no element appears in its original position.",
            "formula": "!n = n! (1 - 1/1! + 1/2! - ... + (-1)^n / n!)"
        },
        {
            "principle": "Pascal's Rule",
            "description": "Each binomial coefficient is the sum of the two above it in Pascal's triangle.",
            "formula": "C(n, k) = C(n - 1, k - 1) + C(n - 1, k)"
        },
        {
            "principle": "Catalan Numbers",
            "description": "Count balanced parentheses, binary trees with n nodes, and monotonic lattice paths that stay on one side of the diagonal.",
            "formula": "C_n = C(2n, n) / (n + 1)"
        },
        {
            "principle": "Stirling Numbers of the Second Kind",
            "description": "Number of ways to partition n labeled items into k non-empty unlabeled subsets.",
            "formula": "S(n, k) = k S(n - 1, k) + S(n - 1, k - 1)"
        }
    ]
}