{"abstract":"One arrangement is mistaken for all arrangements with k successes.","category":"Combinatorics","checks":4,"contract":"Nonnegative integer counts. Permutation and combination counts are zero when a requested selection exceeds the available objects. Shape sizes satisfy the preconditions stated below. Binomial probability rational. Exact operational definition: str(Fraction(math.comb(n,k),2**n))","evaluation_group":"model-6fa373c07ec51f5f","failed_approach":"The denominator counts trials rather than equiprobable outcomes.","family":"xn-binomial-probability-rational","id":"FA-5696","implementations":{"attempt":{"sha256":"7a477dadcee40900eea0e1529e059979898f04f285aaed9fd49f9ddad7e2c9cb","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport calendar\nimport statistics\nimport itertools\nfrom fractions import Fraction\nfrom datetime import date, datetime, timedelta, timezone\nfrom decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR\n\nN = 1\nobservations = []\ndef solve(n, k):\n    return str(Fraction(math.comb(n,k),n)) if n else \"1\"\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (4, 2)', solve(*(4, 2)), '3/8')\ncheck('fixture 2: (3, 0)', solve(*(3, 0)), '1/8')\ncheck('fixture 3: (0, 0)', solve(*(0, 0)), '1')\ncheck('fixture 4: (2, 3)', solve(*(2, 3)), '0')\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"a9b001274ac7c733b2be36b415d2133ac2a4f7435f16bdd5156df962ebeface5","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport calendar\nimport statistics\nimport itertools\nfrom fractions import Fraction\nfrom datetime import date, datetime, timedelta, timezone\nfrom decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR\n\nN = 1\nobservations = []\ndef solve(n, k):\n    return str(Fraction(1,2**n))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (4, 2)', solve(*(4, 2)), '3/8')\ncheck('fixture 2: (3, 0)', solve(*(3, 0)), '1/8')\ncheck('fixture 3: (0, 0)', solve(*(0, 0)), '1')\ncheck('fixture 4: (2, 3)', solve(*(2, 3)), '0')\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"fixed":{"sha256":"1cbba233aa3e29320180dd131e7b64acbbe71d0f6544ce8fc0a2c5074e04ec6b","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\nimport math\nimport calendar\nimport statistics\nimport itertools\nfrom fractions import Fraction\nfrom datetime import date, datetime, timedelta, timezone\nfrom decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR\n\nN = 1\nobservations = []\ndef solve(n, k):\n    return str(Fraction(math.comb(n,k),2**n))\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (4, 2)', solve(*(4, 2)), '3/8')\ncheck('fixture 2: (3, 0)', solve(*(3, 0)), '1/8')\ncheck('fixture 3: (0, 0)', solve(*(0, 0)), '1')\ncheck('fixture 4: (2, 3)', solve(*(2, 3)), '0')\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":" This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"xn-binomial-probability-rational","generated_at":"2026-09-29T14:37:52.689602+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Combinatorics results depend on the stated convention.","repair":"Apply the specified mathematical contract directly, preserving all terms and boundary cases: return str(Fraction(math.comb(n,k),2**n))","root_cause":"One arrangement is mistaken for all arrangements with k successes.","sha256":"b8c3eb6bce4077a00dc3e97ecadc65d46aff2f21bc2eb2d0fef167c23568cf7c","title":"Binomial probability rational · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verification":{"attempt":{"elapsed_ms":76.524,"exit_code":1,"observations":[{"actual":"3/2","check":"fixture 1: (4, 2)","expected":"3/8","passed":false},{"actual":"1/3","check":"fixture 2: (3, 0)","expected":"1/8","passed":false},{"actual":"1","check":"fixture 3: (0, 0)","expected":"1","passed":true},{"actual":"0","check":"fixture 4: (2, 3)","expected":"0","passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (4, 2)\", \"actual\": \"3/2\", \"expected\": \"3/8\", \"passed\": false}, {\"check\": \"fixture 2: (3, 0)\", \"actual\": \"1/3\", \"expected\": \"1/8\", \"passed\": false}, {\"check\": \"fixture 3: (0, 0)\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"fixture 4: (2, 3)\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":103.658,"exit_code":1,"observations":[{"actual":"1/16","check":"fixture 1: (4, 2)","expected":"3/8","passed":false},{"actual":"1/8","check":"fixture 2: (3, 0)","expected":"1/8","passed":true},{"actual":"1","check":"fixture 3: (0, 0)","expected":"1","passed":true},{"actual":"1/4","check":"fixture 4: (2, 3)","expected":"0","passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (4, 2)\", \"actual\": \"1/16\", \"expected\": \"3/8\", \"passed\": false}, {\"check\": \"fixture 2: (3, 0)\", \"actual\": \"1/8\", \"expected\": \"1/8\", \"passed\": true}, {\"check\": \"fixture 3: (0, 0)\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"fixture 4: (2, 3)\", \"actual\": \"1/4\", \"expected\": \"0\", \"passed\": false}], \"passed\": false}\n"},"fixed":{"elapsed_ms":60.587,"exit_code":0,"observations":[{"actual":"3/8","check":"fixture 1: (4, 2)","expected":"3/8","passed":true},{"actual":"1/8","check":"fixture 2: (3, 0)","expected":"1/8","passed":true},{"actual":"1","check":"fixture 3: (0, 0)","expected":"1","passed":true},{"actual":"0","check":"fixture 4: (2, 3)","expected":"0","passed":true}],"passed":true,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (4, 2)\", \"actual\": \"3/8\", \"expected\": \"3/8\", \"passed\": true}, {\"check\": \"fixture 2: (3, 0)\", \"actual\": \"1/8\", \"expected\": \"1/8\", \"passed\": true}, {\"check\": \"fixture 3: (0, 0)\", \"actual\": \"1\", \"expected\": \"1\", \"passed\": true}, {\"check\": \"fixture 4: (2, 3)\", \"actual\": \"0\", \"expected\": \"0\", \"passed\": true}], \"passed\": true}\n"}},"verified":true,"visibility":"public"}