# Question 30849: Candidate Selection Length, Typing, and Distinctness [Return to catalogue page 155](https://ocme.evemisslab.com/en/questions/page/155/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261011-advance-20261015-r06-permutation-combination-foundations/#q-20261011-advance-20261015-r06-permutation-combination-0049) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0060-30849 · q-20261011-advance-20261015-r06-permutation-combination-0049 ## Original question and givens 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, 1], [0], [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, 1], [0], [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. - [mko-permutation-combination](https://ocme.evemisslab.com/en/mko/mko-permutation-combination/) ## Step-by-step derivation S001 · impliesBefore source roles = [qf-density15-selection-fixed-label-qualification, density15_selection_fixed_label_qualification, finite_distinct_selection_protocol] After derived = [[length_flags, typed_flags, distinct_flags, full_flags, ordered_tuples_or_invalid, sorted_subsets_or_invalid]] Conditions: none 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. Step identifiers and dependenciesRule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001 S002 · impliesBefore source roles = [8, 2, 0, [0, 1]] After derived = [Yes, Yes, Yes, Yes, [0, 1], [0, 1]] 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. Step identifiers and dependenciesRule: Selection_EACH_candidate_LENGTH_TYPE_DISTINCT_axes_before_full_guard · Uses: G001, S001 S003 · impliesBefore source roles = [8, 2, 1, [0]] After 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. Step identifiers and dependenciesRule: Selection_EACH_candidate_LENGTH_TYPE_DISTINCT_axes_before_full_guard · Uses: G001, S002 S004 · impliesBefore source roles = [8, 2, 2, [0]] After 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. Step identifiers and dependenciesRule: Selection_EACH_candidate_LENGTH_TYPE_DISTINCT_axes_before_full_guard · Uses: G001, S003 ## Reverse checks - 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, No, No], [Yes, Yes, Yes], [Yes, Yes, Yes], [Yes, No, No], [[0, 1], invalid-selection, invalid-selection], [[0, 1], invalid-selection, invalid-selection]) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 1ab4d0c1127f383121d82885ad83b77af3a139aa281cc57ff4f84b4c67f88a8b - candidate record SHA-256: 6fbfc3df27ad7f4a57bcd9e4f8f0a20b1dac71eb4019bf0ffe52dc1de2d0a71a - semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf - question-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261011-advance-20261015-r06-permutation-combination-0049.json)