Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 8], [0, 1], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.
G001 — Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. Labels=Fin(8)={0,…,7}, k=2, candidates=[[0, 8], [0, 1], [0]]. Give (length mask, all-labels-in-domain mask, all-distinct mask, full eligibility mask, eligible ordered tuples or invalid-selection, eligible sorted subsets or invalid-selection), preserving candidate order.
Definition and scope
Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.
density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status.
derived = [Yes, No, Yes, No, invalid selection, invalid selection]
Conditions: none
Wrong-length,outside-domain or repeated-label RAW candidates are legal No cases.Check LENGTH,TYPE andDISTINCTNESS separately,then conjoin.ONLY a good candidate yields its ORIGINAL ordered row and separately sorted subset.At k0 an empty row is valid even in Fin0;retain all three candidate positions.
Wrong-length,outside-domain or repeated-label RAW candidates are legal No cases.Check LENGTH,TYPE andDISTINCTNESS separately,then conjoin.ONLY a good candidate yields its ORIGINAL ordered row and separately sorted subset.At k0 an empty row is valid even in Fin0;retain all three candidate positions.
derived = [No, Yes, Yes, No, invalid selection, invalid selection]
Conditions: none
Wrong-length,outside-domain or repeated-label RAW candidates are legal No cases.Check LENGTH,TYPE andDISTINCTNESS separately,then conjoin.ONLY a good candidate yields its ORIGINAL ordered row and separately sorted subset.At k0 an empty row is valid even in Fin0;retain all three candidate positions.
S004 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S003 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S002 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S001 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.
This question review and source binding
Review date: 2026-10-26
Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.