FA-5741 / Combinatorics / Open access
Cumulative binomial count · case 01
An exact-cardinality count replaces a cumulative count.
ROOT CAUSE
An exact-cardinality count replaces a cumulative count.
VERIFIED REPAIR
Apply the specified mathematical contract directly, preserving all terms and boundary cases: return sum(math.comb(n,i) for i in range(min(n,k)+1))
Unsuccessful approach: The allowed maximum cardinality is excluded.
Case 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. Cumulative binomial count. Exact operational definition: sum(math.comb(n,i) for i in range(min(n,k)+1))
Why this case matters
Small exact fixtures expose this error without platform timing, external services, or probabilistic observations. Combinatorics results depend on the stated convention.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR
N = 1
observations = []
def solve(n, k):
return math.comb(n,k)
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (4, 2)', solve(*(4, 2)), 11)
check('fixture 2: (3, 0)', solve(*(3, 0)), 1)
check('fixture 3: (2, 5)', solve(*(2, 5)), 4)
check('fixture 4: (0, 0)', solve(*(0, 0)), 1)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| fixture 1: (4, 2) | 6 | 11 | Failed |
| fixture 2: (3, 0) | 1 | 1 | Passed |
| fixture 3: (2, 5) | 0 | 4 | Failed |
| fixture 4: (0, 0) | 1 | 1 | Passed |
SHA-256 / 49f4655c63edaa5864f7dd82270bf8675136e1293f17d18f4e4892008b87d3fd
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR
N = 1
observations = []
def solve(n, k):
return sum(math.comb(n,i) for i in range(min(n,k)))
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (4, 2)', solve(*(4, 2)), 11)
check('fixture 2: (3, 0)', solve(*(3, 0)), 1)
check('fixture 3: (2, 5)', solve(*(2, 5)), 4)
check('fixture 4: (0, 0)', solve(*(0, 0)), 1)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| fixture 1: (4, 2) | 5 | 11 | Failed |
| fixture 2: (3, 0) | 0 | 1 | Failed |
| fixture 3: (2, 5) | 3 | 4 | Failed |
| fixture 4: (0, 0) | 0 | 1 | Failed |
SHA-256 / 1553e3b72de69fb02d9b866b1e079f83677aa0ad2e1a298c54624cf2e19c1022
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
import calendar
import statistics
import itertools
from fractions import Fraction
from datetime import date, datetime, timedelta, timezone
from decimal import Decimal, ROUND_HALF_UP, ROUND_DOWN, ROUND_CEILING, ROUND_FLOOR
N = 1
observations = []
def solve(n, k):
return sum(math.comb(n,i) for i in range(min(n,k)+1))
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
check('fixture 1: (4, 2)', solve(*(4, 2)), 11)
check('fixture 2: (3, 0)', solve(*(3, 0)), 1)
check('fixture 3: (2, 5)', solve(*(2, 5)), 4)
check('fixture 4: (0, 0)', solve(*(0, 0)), 1)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| fixture 1: (4, 2) | 11 | 11 | Passed |
| fixture 2: (3, 0) | 1 | 1 | Passed |
| fixture 3: (2, 5) | 4 | 4 | Passed |
| fixture 4: (0, 0) | 1 | 1 | Passed |
SHA-256 / e6cd62a55b8b0f37d498466248af9202362f91cd6b286150ba69144b4866ad92
Verification & scope
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.
Observations recorded using Python 3.12.14 at 2026-09-29T14:37:53.093521+00:00.
Case digest / 14c3a984673986d17b2f8c743473d3e54ba36ce8d574795adbde3961c874da97