{"abstract":"Distinct-subset counting forbids permitted repeats.","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. Multiset selection count. Exact operational definition: math.comb(n+k-1,k) if n else int(k==0)","contract_signature":"n, k","evaluation_group":"model-1724b0479f1a837e","failed_approach":"Ordered tuples distinguish arrangements that represent the same multiset.","family":"xn-multiset-selection-count","id":"FA-5601","implementations":{"attempt":{"sha256":"311610e6694cf5157e5ebaa394128ad5c28f1ed428cb328fcf5e28f7c119da49","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 n**k\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (3, 2)', solve(*(3, 2)), 6)\ncheck('fixture 2: (2, 3)', solve(*(2, 3)), 4)\ncheck('fixture 3: (0, 0)', solve(*(0, 0)), 1)\ncheck('fixture 4: (0, 2)', solve(*(0, 2)), 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":"4a39ea83b94b72d64a977dd239e200bc0e8f4378ef21b170ae545104b240b270","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 math.comb(n,k)\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\ncheck('fixture 1: (3, 2)', solve(*(3, 2)), 6)\ncheck('fixture 2: (2, 3)', solve(*(2, 3)), 4)\ncheck('fixture 3: (0, 0)', solve(*(0, 0)), 1)\ncheck('fixture 4: (0, 2)', solve(*(0, 2)), 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-multiset-selection-count","generated_at":"2026-09-29T14:37:51.270969+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.","root_cause":"Distinct-subset counting forbids permitted repeats.","sha256":"4fc1fbaceb0b8c1facf96425320062c80349b9750d6d36fafcf51e573e4d326f","title":"Multiset selection count · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verified":true,"visibility":"public","verification":{"attempt":{"elapsed_ms":47.0,"exit_code":1,"observations":[{"actual":9,"check":"fixture 1: (3, 2)","expected":6,"passed":false},{"actual":8,"check":"fixture 2: (2, 3)","expected":4,"passed":false},{"actual":1,"check":"fixture 3: (0, 0)","expected":1,"passed":true},{"actual":0,"check":"fixture 4: (0, 2)","expected":0,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (3, 2)\", \"actual\": 9, \"expected\": 6, \"passed\": false}, {\"check\": \"fixture 2: (2, 3)\", \"actual\": 8, \"expected\": 4, \"passed\": false}, {\"check\": \"fixture 3: (0, 0)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 4: (0, 2)\", \"actual\": 0, \"expected\": 0, \"passed\": true}], \"passed\": false}\n"},"broken":{"elapsed_ms":49.967,"exit_code":1,"observations":[{"actual":3,"check":"fixture 1: (3, 2)","expected":6,"passed":false},{"actual":0,"check":"fixture 2: (2, 3)","expected":4,"passed":false},{"actual":1,"check":"fixture 3: (0, 0)","expected":1,"passed":true},{"actual":0,"check":"fixture 4: (0, 2)","expected":0,"passed":true}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"fixture 1: (3, 2)\", \"actual\": 3, \"expected\": 6, \"passed\": false}, {\"check\": \"fixture 2: (2, 3)\", \"actual\": 0, \"expected\": 4, \"passed\": false}, {\"check\": \"fixture 3: (0, 0)\", \"actual\": 1, \"expected\": 1, \"passed\": true}, {\"check\": \"fixture 4: (0, 2)\", \"actual\": 0, \"expected\": 0, \"passed\": true}], \"passed\": false}\n"}},"member_only":{"stages":["fixed"],"fields":["implementations.fixed","verification.fixed","harness","repair"],"note":"The verified repair, its recorded checks, the repair description, and the scoring harness are available to members."}}