Source: https://guthlabs.ai/docs/instruments/combinatorics-calculator-ncr-npr-multinomial/

instrument / verified-clean

# Combinatorics Calculator (nCr/nPr/multinomial)

The current ledger marks this instrument verified clean within the source-bound verification corpus.

Status: **VERIFIED_CLEAN**Lifecycle: **verified-clean**

## Inspectable records

[01**Overview**Purpose and lifecycle](<https://guthlabs.ai/docs/instruments/combinatorics-calculator-ncr-npr-multinomial/>)[02**Mechanism**Method and assumptions](<https://guthlabs.ai/docs/instruments/combinatorics-calculator-ncr-npr-multinomial/mechanism/>)[03**Implementation**Binding and validation](<https://guthlabs.ai/docs/instruments/combinatorics-calculator-ncr-npr-multinomial/implementation/>)[04**Qualification**Evidence and errata](<https://guthlabs.ai/docs/instruments/combinatorics-calculator-ncr-npr-multinomial/qualification/>)

## Ledger state

The current ledger marks this instrument verified clean within the source-bound verification corpus.

**Area**: Mathematics — arithmetic through advanced (algebra, matrices, calculus, sequences/series, number theory, combinatorics, graphing, complex numbers, geometry, logic, sets)

**Tranche**: tranche-6

**Evidence ID**: t6-066

**Ledger rows**: 1

No ledger note is attached.

## Review findings

The three supplied formulas are standard and pass the four checks above, including the n=0,r=0 boundary. The overall specification cannot be accepted as complete because it claims permutations with repetition, Catalan numbers, and Stirling numbers without defining their meanings or algorithms, and it omits essential input constraints.

### Formula and scope findings

- The stated formulas correctly define ordinary combinations, ordinary permutations, and multinomial coefficients, subject to integer-domain constraints.

- 'Permutations with repetition' is not defined and is ambiguous: it may mean ordered selections with replacement (n^r) or permutations of a multiset (n!/(k1!k2!...)).

- Catalan numbers are claimed but no Catalan formula or algorithm is provided; a standard formula is C_n=(1/(n+1))·binomial(2n,n).

- Stirling numbers are claimed but no type (first or second kind), recurrence, or formula is provided.

- The multinomial formula requires nonnegative integer parts k1,...,km whose sum equals n; this required constraint is not stated.

- nCr requires integers n>=0 and 0<=r<=n; nPr requires integers n>=0 and 0<=r<=n. These domain constraints are omitted.

### Unit assumptions

- n, r, and every partition entry k_i are dimensionless, nonnegative integers.

- Factorials are defined only for nonnegative integers, with 0!=1.

- For a multinomial input given only as a partition list, n is assumed to be the exact sum of the list entries.

- Outputs are exact integer counts, not floating-point approximations.

## Boundary hardening

**BREAK_CLAIMED** · SELF_FLAGGED

The specification exposes an undefined factorial domain at r>n and does not state required rejection behavior. It also lists permutations with repetition without giving its formula or an explicit convention selector, allowing the same nPr-looking input to produce different results.

## Errata and corrections

No applied erratum is attached to this unique-name record.

## Inspectable example

4 review vectors are recorded. One source-bound sample is shown below.

````
{
  "inputs": {
    "operation": "nCr",
    "n": 0,
    "r": 0
  },
  "expected_outputs": {
    "coefficient": 1
  },
  "working": "0!/(0!·(0-0)!) = 0!/(0!·0!) = 1/(1·1) = 1. This checks the zero boundary and uses 0!=1."
}
````

## Related workflows

No source-defined multi-instrument workflow membership is recorded.

## Sources and evidence

These pointers bind the page to evidence. They do not grant authority or collapse verification into deployment.

- `VERIFIED_LEDGER.json · VERIFIED_CLEAN`

- `evidence-tranche6/tranche6-results.json`

- `BOUNDARY_HAZARD_CATALOG.json`