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, 0], [0, 8], [0, 1]]. 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, 0], [0, 8], [0, 1]]. 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, Yes, No, 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.
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.
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.
Result
([Yes, Yes, Yes], [Yes, No, Yes], [No, Yes, Yes], [No, No, Yes], [invalid-selection, invalid-selection, [0, 1]], [invalid-selection, invalid-selection, [0, 1]])
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.