{ "model": { "rank": 8553, "code": "thing-q209875", "model_id": "vr.tr.binomial-coefficient", "name": "binomial coefficient", "purpose": "Let an agent compute and explain binomial coefficients, their identities and applications in combinatorics, probability and algebra, and avoid overflow and off-by-one errors.", "family": "Thing Registry", "category": "Cross-cutting context", "status": "research-draft", "kind": "thing", "plane": "XCT", "domain": "XCT.QTY", "industry": "", "version": "", "url": "/models/thing/q209875/", "tier": 2, "score": 50, "payload": { "layer": "wikidata", "aliases": [ "central binomial coefficient" ], "aliasCount": 1, "merged": 1, "knownIn": 50, "facets": null, "markers": [], "lexicalClass": "", "senseRank": null, "alsoRegisteredAs": null, "source": { "dataset": "wikidata", "item": "Q209875", "url": "https://www.wikidata.org/wiki/Q209875", "license": "CC0 1.0" } }, "research": { "vercy": "1.0-draft", "publication": { "status": "research-draft", "adjudicationStatus": "unreviewed", "publishableCanonical": false, "generatedAt": "2026-09-11T15:32:53Z", "providers": [ "Claude" ], "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify", "pass": 2, "wave": 1, "engine": "claude" }, "metaModel": { "id": "THING-Q209875", "registryId": "vr.tr.binomial-coefficient", "name": "binomial coefficient", "version": "0.2.0-wave.1", "entryKind": "thing", "family": "Thing Registry", "domain": [ "XCT.QTY" ], "status": "research-draft" }, "canonicalUrl": "https://ver.cy/models/thing/q209875/", "model": { "registry_id": "vr.tr.binomial-coefficient", "name": "binomial coefficient", "purpose": "Let an agent compute and explain binomial coefficients, their identities and applications in combinatorics, probability and algebra, and avoid overflow and off-by-one errors.", "definition": "The number of ways to choose k items from n without regard to order, written C(n, k) or n choose k, equal to n! divided by k!(n-k)!, appearing as coefficients in the binomial expansion of (x + y)^n and as entries of Pascal triangle; central binomial coefficients are the middle entries C(2n, n).", "what_it_is_for": "Counting combinations and expanding binomials.", "affordances": [ "compute C(n, k) exactly", "explain identities and Pascal triangle", "apply to probability and expansions", "compute large values safely" ], "distinguishing_features": [ "Non-negative integer", "Symmetric in k and n-k", "Pascal rule recurrence", "Ubiquitous in mathematics" ], "appearance": "Not physical; numbers in formulas and Pascal triangle.", "visual_identification": [ "Notation n over k in parentheses", "Rows of Pascal triangle", "Permutations count ordered selections instead" ], "physical_properties": [], "families_and_kinds": [ "ordinary binomial coefficients", "central binomial coefficients", "generalised binomial coefficients", "multinomial coefficients (related)" ], "related_models": [ { "relation": "is a kind of", "target": "non-negative integer", "why": "category" }, { "relation": "appears in", "target": "binomial theorem", "why": "theorem" }, { "relation": "is related to", "target": "theorem", "why": "mathematical results" }, { "relation": "is related to", "target": "mathematics", "why": "field" } ], "identifiers": [], "standards_and_regulation": [ "ISO 80000-2 mathematical notation" ], "failure_modes_and_hazards": [ "Integer overflow", "Confusing combinations with permutations", "Off-by-one errors" ], "in_scope": [], "out_of_scope": [], "characteristics": [] }, "sources": [], "structure": { "bundles": [ { "id": "compute", "name": "Compute", "description": "Calculating values.", "rationale": "Exact methods exist.", "layers": [ { "id": "value", "name": "Value", "description": "C(n, k).", "findings": [ { "id": "value-finding", "name": "Value", "description": "Value.", "questions": [ { "text": "What is C(n, k) for these values?", "kind": "measurement" }, { "text": "Which method avoids overflow, such as multiplicative or Pascal recurrence?", "kind": "action" } ] } ] }, { "id": "large", "name": "Large", "description": "Large arguments.", "findings": [ { "id": "large-finding", "name": "Large", "description": "Large arguments.", "questions": [ { "text": "How can large binomial coefficients be computed or approximated?", "kind": "action" }, { "text": "Is a modular or logarithmic value sufficient?", "kind": "boundary" } ] } ] } ] }, { "id": "identities", "name": "Identities", "description": "Properties.", "rationale": "Identities simplify work.", "layers": [ { "id": "pascal", "name": "Pascal", "description": "Pascal rule.", "findings": [ { "id": "pascal-finding", "name": "Pascal", "description": "Pascal rule.", "questions": [ { "text": "How does the Pascal rule build the triangle?", "kind": "definition" }, { "text": "What symmetry holds?", "kind": "definition" } ] } ] }, { "id": "sums", "name": "Sums", "description": "Summation identities.", "findings": [ { "id": "sums-finding", "name": "Sums", "description": "Sums.", "questions": [ { "text": "What is the sum of a row, and what other identities apply?", "kind": "definition" }, { "text": "How is Vandermonde identity used?", "kind": "definition" } ] } ] } ] }, { "id": "apply", "name": "Apply", "description": "Uses.", "rationale": "Combinatorics and probability.", "layers": [ { "id": "probability", "name": "Probability", "description": "Binomial distribution.", "findings": [ { "id": "probability-finding", "name": "Probability", "description": "Probability.", "questions": [ { "text": "How do binomial coefficients give binomial probabilities?", "kind": "definition" }, { "text": "What is the probability of k successes in n trials?", "kind": "measurement" } ] } ] }, { "id": "algebra", "name": "Algebra", "description": "Expansions.", "findings": [ { "id": "algebra-finding", "name": "Algebra", "description": "Expansions.", "questions": [ { "text": "What are the coefficients of this binomial expansion?", "kind": "measurement" }, { "text": "How do they generalise to multinomials?", "kind": "definition" } ] } ] } ] }, { "id": "learn", "name": "Learn", "description": "Teaching.", "rationale": "Common confusions.", "layers": [ { "id": "versus", "name": "Versus", "description": "Combinations and permutations.", "findings": [ { "id": "versus-finding", "name": "Versus", "description": "Combinations versus permutations.", "questions": [ { "text": "Does this problem need combinations or permutations?", "kind": "boundary" }, { "text": "Does order matter?", "kind": "boundary" } ] } ] }, { "id": "central", "name": "Central", "description": "Central coefficients.", "findings": [ { "id": "central-finding", "name": "Central", "description": "Central coefficients.", "questions": [ { "text": "What are central binomial coefficients, and where do they appear?", "kind": "definition" }, { "text": "How fast do they grow?", "kind": "measurement" } ] } ] } ] } ] }, "openQuestions": [ "Should Pascal triangle be a separate entry?", "How should computational libraries be linked?", "How should generalisations be linked?" ], "statistics": { "bundles": 4, "layers": 8, "findings": 8, "questions": 16 } }, "draft": { "generator": "vr.draft.v3", "status": "draft-generated", "researched": false, "archetype": "unit, quantity or measure", "method": "Written from the archetype playbook - what this kind of thing needs beyond identity and provenance - and from the structure that recurred across 6,333 models already researched by two engines. Applied to this entry by rule. No source was read for this thing and no claim here is researched. This entry carries no facets of its own, so they were inferred from its domain - a guess about a whole domain applied to one thing.", "facetsInferred": true, "nextPass": "A researcher replaces this draft with a sourced specification. Treat every sentence below as a proposal to argue with.", "purpose": "Give an agent a durable, checkable way to recognise a binomial coefficient, record what state it is in, and decide what may be done with it.", "whatItIs": "Let an agent compute and explain binomial coefficients, their identities and applications in combinatorics, probability and algebra, and avoid overflow and off-by-one errors.", "characteristics": { "substance": "abstract", "origin": "conceptual", "agency": "inert" }, "whatYouCanDoWithIt": [ "observed and measured" ], "distinguishingFeatures": [ "Names folded into this entry, which a task may need to split apart again: central binomial coefficient.", "1 finer distinctions are held as aliases rather than separate entries, because telling them apart needs a task that asks for it.", "Described in 50 Wikipedia languages, which is a measure of how widely the thing is known, not of how important it is." ], "openQuestionsForResearch": [ "Which of the bundles below does a real task actually need, and which are ceremony?", "What does this thing have that the facets do not capture at all?", "Which neighbouring kind is most often confused with a binomial coefficient, and on what evidence are they told apart?" ], "whatItIsMadeOf": "an abstraction with no physical instance", "physicalCharacter": [ "Does nothing on its own; everything it does, something else did to it.", "These come from the domain this entry sits in rather than from the entry itself, so treat them as a first guess about the whole domain applied to one thing." ], "whatCanBeDoneWithIt": [ "observe it, measure it, record its state" ], "howItIsRecognised": [ "Nothing to see: a unit is a convention. What is seen is an instrument reading in it." ], "relatedModels": [ { "relation": "covers", "note": "Finer kinds folded into this entry because telling them apart needs a task that asks for it. Each is a model waiting to be split out when one does.", "targets": [ "central binomial coefficient" ] } ], "standing": "Described in 50 Wikipedia languages, which measures how widely it is written about rather than how important or how common it is. 1 finer distinctions are held inside this entry as names rather than as separate models.", "structure": { "bundles": [ { "id": "identity-and-classification", "name": "Identity, naming and classification", "description": "How an agent tells one binomial coefficient from another, and a binomial coefficient from things that resemble it.", "rationale": "Recognition comes before every other claim. Without stable identity nothing else in the model can be trusted to be about the same thing twice.", "layers": [ { "id": "naming-and-identifiers", "name": "Names and identifiers", "description": "The names this thing goes by and the identifiers that survive translation and time.", "findings": [ { "id": "preferred-name-and-aliases", "name": "Preferred name, aliases and local names", "description": "Which name to use, which names mean the same thing, and which merely sound similar.", "questions": [ { "id": "preferred-name-and-aliases-q01", "text": "What identifies and describes the name of a binomial coefficient, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "preferred-name-and-aliases-q02", "text": "Who or what asserted this about the name of a binomial coefficient, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "preferred-name-and-aliases-q03", "text": "What may an agent decide or do once the name of a binomial coefficient is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "stable-identifiers", "name": "Stable identifiers and external keys", "description": "Identifiers that keep pointing at this kind of thing across systems and languages.", "questions": [ { "id": "stable-identifiers-q01", "text": "What identifies and describes an identifier for a binomial coefficient, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stable-identifiers-q02", "text": "Who or what asserted this about an identifier for a binomial coefficient, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stable-identifiers-q03", "text": "What may an agent decide or do once an identifier for a binomial coefficient is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "classification-and-granularity", "name": "Classification and granularity", "description": "Where a binomial coefficient sits among kinds, and how finely a task needs to cut it.", "findings": [ { "id": "kind-and-parents", "name": "Kind, parents and neighbouring kinds", "description": "The classes this thing belongs to and the ones it is next to.", "questions": [ { "id": "kind-and-parents-q01", "text": "What identifies and describes the kind of a binomial coefficient, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "kind-and-parents-q02", "text": "Who or what asserted this about the kind of a binomial coefficient, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "kind-and-parents-q03", "text": "What may an agent decide or do once the kind of a binomial coefficient is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] }, { "id": "distinguishing-features", "name": "Distinguishing features", "description": "What separates a binomial coefficient from the things most often confused with it.", "questions": [ { "id": "distinguishing-features-q01", "text": "What identifies and describes what distinguishes a binomial coefficient, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "distinguishing-features-q02", "text": "Who or what asserted this about what distinguishes a binomial coefficient, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "distinguishing-features-q03", "text": "What may an agent decide or do once what distinguishes a binomial coefficient is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "state-and-lifecycle", "name": "State and lifecycle", "description": "The states a binomial coefficient passes through and the events that move it between them.", "rationale": "Most decisions about a thing depend on what state it is in now, which is a claim with a time on it, not a property.", "layers": [ { "id": "lifecycle-stages", "name": "Lifecycle stages", "description": "From coming into existence to ceasing to be one of these.", "findings": [ { "id": "stages-and-transitions", "name": "Stages and transitions", "description": "The stages worth naming and what moves a binomial coefficient between them.", "questions": [ { "id": "stages-and-transitions-q01", "text": "What identifies and describes the lifecycle of a binomial coefficient, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "stages-and-transitions-q02", "text": "Who or what asserted this about the lifecycle of a binomial coefficient, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "stages-and-transitions-q03", "text": "What may an agent decide or do once the lifecycle of a binomial coefficient is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "observations-and-status", "name": "Observations and current status", "description": "What is observed about a binomial coefficient, how often and by whom.", "findings": [ { "id": "observation-record", "name": "Observation record", "description": "How an observation of a binomial coefficient is recorded so that it can be superseded rather than overwritten.", "questions": [ { "id": "observation-record-q01", "text": "What identifies and describes an observation of a binomial coefficient, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "observation-record-q02", "text": "Who or what asserted this about an observation of a binomial coefficient, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "observation-record-q03", "text": "What may an agent decide or do once an observation of a binomial coefficient is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] }, { "id": "definition-and-realisation", "name": "Definition, dimension and realisation", "description": "What binomial coefficient is defined as and how it is realised in practice.", "rationale": "A unit is not the quantity it measures, and its definition has a history that older data still depends on.", "layers": [ { "id": "definition", "name": "Definition and dimension", "description": "The defining relation and the quantity it measures.", "findings": [ { "id": "definition-record", "name": "Definition, body and date", "description": "What defines it now, and what defined it before.", "questions": [ { "id": "definition-record-q01", "text": "How is binomial coefficient defined, by which body, and since when?", "kind": "definition" }, { "id": "definition-record-q02", "text": "What was the previous definition, and what data still depends on it?", "kind": "provenance" } ] } ] }, { "id": "conversion", "name": "Conversion and compatibility", "description": "Relations to other units, exact or approximate.", "findings": [ { "id": "conversion-record", "name": "Conversions and their exactness", "description": "Factors, and whether each is exact by definition.", "questions": [ { "id": "conversion-record-q01", "text": "What conversions relate binomial coefficient to other units, and which are exact by definition?", "kind": "measurement" }, { "id": "conversion-record-q02", "text": "Where does rounding a conversion change a decision, and how should that be recorded?", "kind": "action" } ] } ] } ] }, { "id": "provenance-and-evidence", "name": "Provenance, evidence and time", "description": "Where every claim about a binomial coefficient came from and when it held.", "rationale": "A claim without a source and a time cannot be superseded, only overwritten, and an agent that overwrites loses the ability to explain itself.", "layers": [ { "id": "source-and-authority", "name": "Source and authority", "description": "Who said it, on what evidence, and how strongly.", "findings": [ { "id": "claim-provenance", "name": "Claim provenance and confidence", "description": "The authority behind each claim about a binomial coefficient and how confident it is.", "questions": [ { "id": "claim-provenance-q01", "text": "What identifies and describes a claim about a binomial coefficient, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "claim-provenance-q02", "text": "Who or what asserted this about a claim about a binomial coefficient, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "claim-provenance-q03", "text": "What may an agent decide or do once a claim about a binomial coefficient is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] }, { "id": "time-and-versions", "name": "Time, versions and supersession", "description": "When a claim was true, when it was learnt, and what replaced it.", "findings": [ { "id": "validity-and-supersession", "name": "Validity period and supersession", "description": "How an old claim about a binomial coefficient is retired without being erased.", "questions": [ { "id": "validity-and-supersession-q01", "text": "What identifies and describes the validity of a claim about a binomial coefficient, and in what units or vocabulary?", "kind": "definition", "answer_data": [ "identifiers", "types and classes", "values with units", "explicit unknowns" ] }, { "id": "validity-and-supersession-q02", "text": "Who or what asserted this about the validity of a claim about a binomial coefficient, by which method, and when was it true?", "kind": "provenance", "answer_data": [ "authority", "method", "evidence", "event time", "knowledge time" ] }, { "id": "validity-and-supersession-q03", "text": "What may an agent decide or do once the validity of a claim about a binomial coefficient is known, and what must it refuse?", "kind": "action", "answer_data": [ "permitted actions", "preconditions", "refusals", "escalation" ] } ] } ] } ] } ] }, "statistics": { "bundles": 4, "layers": 8, "findings": 10, "questions": 28 } } } }