# Question 30851: Completions after Fixing an Initial Ordered Prefix [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-0076) 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. R0060-30851 · q-20261011-advance-20261015-r06-permutation-combination-0076 ## 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. n=0, k=0, r=0; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio). - 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. n=0, k=0, r=0; the first r positions use labels 0,…,r−1 in that fixed order, empty if r=0. Give (remaining objects, remaining positions, completion count, full ordered count, completion/full count ratio). ## 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-forced-initial-order, density15_selection_forced_initial_order, finite_distinct_selection_protocol] After derived = [[remaining_objects, remaining_positions, completions, full_ordered_count, COUNT_ratio]] 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 · calculationBefore Selection prefix ORIGINAL n minus used r := sub(0, 0) After Selection prefix ORIGINAL n minus used r = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Selection prefix ORIGINAL k minus filled r := sub(0, 0) After Selection prefix ORIGINAL k minus filled r = 0 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · impliesBefore source roles = [Selection prefix remaining ordered, 0, 0] After derived = [[], 1] Conditions: none At position j there are n-j UNUSED distinct labeled objects.Multiply ALL k available-choice counts;at k0 there is exactly ONE empty ordered tuple.This is not replacement,circular or indistinguishable selection. Step identifiers and dependenciesRule: Selection_ORDERED_distinct_without_replacement_available_choices · Uses: G001, S003 S005 · impliesBefore source roles = [Selection full ordered, 0, 0] After derived = [[], 1] Conditions: none At position j there are n-j UNUSED distinct labeled objects.Multiply ALL k available-choice counts;at k0 there is exactly ONE empty ordered tuple.This is not replacement,circular or indistinguishable selection. Step identifiers and dependenciesRule: Selection_ORDERED_distinct_without_replacement_available_choices · Uses: G001, S004 S006 · calculationBefore Selection ORIGINAL completion COUNT ratio := div(1, 1) After Selection ORIGINAL completion COUNT ratio = 1 Conditions: none Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · impliesBefore source roles = [0, 0, 0, []] After derived = [0, 0, 1, 1, 1/1] Conditions: none The FIRST r positions use labels0..r-1 in their GIVEN order.Subtract used objects/filled positions first.At r0 retain the full count;at r=k there is one completion.The final value is a COUNT ratio,not a probability without a uniform law. Step identifiers and dependenciesRule: Selection_FIXED_prefix_labels_and_COUNT_ratio_NOT_probability · Uses: G001, S006 ## Reverse checks - S007 · 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. - S006 · premise_and_result_audit — Recompute Selection ORIGINAL completion COUNT ratio with exact rational arithmetic, not rounded decimals. - S005 · 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. - 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 — Recompute Selection prefix ORIGINAL k minus filled r with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Selection prefix ORIGINAL n minus used r with exact rational arithmetic, not rounded decimals. - 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 (0, 0, 1, 1, 1/1) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 3bc2137a06b26378b2e959ceb9399f6983e198fe368f91261da814adee1626ca - candidate record SHA-256: 86d2fde4e75f4c7f0054163b8fdaeafab5c7d4a752637eb8ab7b3fb14aa49d6f - 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-0076.json)