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  "passed": 11,
  "failed": 0,
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      "passed": 2
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    "boundary": {
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    "property": {
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  "internally_reviewed": false,
  "review_kind": null,
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  "reviewer_role": null,
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      "sources",
      "tests"
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  "internal_review": {
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  "api": "https://www.calculatorx.com/api/v1/calc/combinatorics",
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  "schema_output": "https://www.calculatorx.com/schemas/math.combinatorics.output.json",
  "assumptions": [
    "Integers 0 ≤ k ≤ n ≤ 1000",
    "Not factorial n!",
    "Not multinomial and not with-replacement"
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  "limitations": [
    "Not the factorial product page — use /calc/math/factorial for n!",
    "Not multinomial coefficients and not with-replacement selections",
    "Domain 0 ≤ k ≤ n ≤ 1000",
    "k > n → INVALID_INPUT",
    "unknown mode → INVALID_MODE"
  ],
  "boundary_conditions": [
    "missing n or k → MISSING_REQUIRED_INPUT",
    "k > n → INVALID_INPUT",
    "n > 1000 → VALUE_ABOVE_MAX",
    "unknown mode → INVALID_MODE"
  ],
  "sources": [
    "ISO 80000-2:2019 Quantities and units — Mathematics (combinatorial numbers)",
    "NIST / standard discrete-math identity P(n,k) = C(n,k)·k!"
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  "fingerprint": "sha256:7e6eeaa69cdb1b2c70587975c2bc20dae683842b8ac1fcb2070fd44859cda07e",
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      "expectWarning": null,
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        "mode": "combination",
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      "expectWarning": null,
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        "n": 10,
        "k": 3,
        "mode": "permutation",
        "symbol": "P(10,3)",
        "formula": "P(n,k) = n! / (n−k)!",
        "value": 720,
        "exact": "720",
        "digits": 3,
        "max_n": 1000
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      "tol": null
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      "id": "boundary-k-gt-n",
      "ok": true,
      "kind": "boundary",
      "inputs": {
        "n": 5,
        "k": 6
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      "expectError": "INVALID_INPUT",
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      "error": "k must be ≤ n."
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    {
      "id": "boundary-combo-missing",
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      "code": "MISSING_REQUIRED_INPUT",
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      "detail": "P(n,n)=n!",
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