# Question 30854: 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-0079) 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. R0060-30854 · q-20261011-advance-20261015-r06-permutation-combination-0079 ## 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=32, k=32, r=16; 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=32, k=32, r=16; 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(32, 16) After Selection prefix ORIGINAL n minus used r = 16 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(32, 16) After Selection prefix ORIGINAL k minus filled r = 16 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 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor0 := mul(1, 16) After Selection prefix remaining ordered ORIGINAL available-choice factor0 = 16 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, S003 S005 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor1 := mul(16, 15) After Selection prefix remaining ordered ORIGINAL available-choice factor1 = 240 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, S004 S006 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor2 := mul(240, 14) After Selection prefix remaining ordered ORIGINAL available-choice factor2 = 3360 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 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor3 := mul(3360, 13) After Selection prefix remaining ordered ORIGINAL available-choice factor3 = 43680 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, S006 S008 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor4 := mul(43680, 12) After Selection prefix remaining ordered ORIGINAL available-choice factor4 = 524160 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, S007 S009 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor5 := mul(524160, 11) After Selection prefix remaining ordered ORIGINAL available-choice factor5 = 5765760 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, S008 S010 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor6 := mul(5765760, 10) After Selection prefix remaining ordered ORIGINAL available-choice factor6 = 57657600 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, S009 S011 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor7 := mul(57657600, 9) After Selection prefix remaining ordered ORIGINAL available-choice factor7 = 518918400 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, S010 S012 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor8 := mul(518918400, 8) After Selection prefix remaining ordered ORIGINAL available-choice factor8 = 4151347200 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, S011 S013 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor9 := mul(4151347200, 7) After Selection prefix remaining ordered ORIGINAL available-choice factor9 = 29059430400 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, S012 S014 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor10 := mul(29059430400, 6) After Selection prefix remaining ordered ORIGINAL available-choice factor10 = 174356582400 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, S013 S015 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor11 := mul(174356582400, 5) After Selection prefix remaining ordered ORIGINAL available-choice factor11 = 871782912000 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, S014 S016 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor12 := mul(871782912000, 4) After Selection prefix remaining ordered ORIGINAL available-choice factor12 = 3487131648000 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, S015 S017 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor13 := mul(3487131648000, 3) After Selection prefix remaining ordered ORIGINAL available-choice factor13 = 10461394944000 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, S016 S018 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor14 := mul(10461394944000, 2) After Selection prefix remaining ordered ORIGINAL available-choice factor14 = 20922789888000 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, S017 S019 · calculationBefore Selection prefix remaining ordered ORIGINAL available-choice factor15 := mul(20922789888000, 1) After Selection prefix remaining ordered ORIGINAL available-choice factor15 = 20922789888000 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, S018 S020 · impliesBefore source roles = [Selection prefix remaining ordered, 16, 16] After derived = [[16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], 20922789888000] 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, S019 S021 · calculationBefore Selection full ordered ORIGINAL available-choice factor0 := mul(1, 32) After Selection full ordered ORIGINAL available-choice factor0 = 32 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, S020 S022 · calculationBefore Selection full ordered ORIGINAL available-choice factor1 := mul(32, 31) After Selection full ordered ORIGINAL available-choice factor1 = 992 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, S021 S023 · calculationBefore Selection full ordered ORIGINAL available-choice factor2 := mul(992, 30) After Selection full ordered ORIGINAL available-choice factor2 = 29760 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, S022 S024 · calculationBefore Selection full ordered ORIGINAL available-choice factor3 := mul(29760, 29) After Selection full ordered ORIGINAL available-choice factor3 = 863040 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, S023 S025 · calculationBefore Selection full ordered ORIGINAL available-choice factor4 := mul(863040, 28) After Selection full ordered ORIGINAL available-choice factor4 = 24165120 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, S024 S026 · calculationBefore Selection full ordered ORIGINAL available-choice factor5 := mul(24165120, 27) After Selection full ordered ORIGINAL available-choice factor5 = 652458240 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, S025 S027 · calculationBefore Selection full ordered ORIGINAL available-choice factor6 := mul(652458240, 26) After Selection full ordered ORIGINAL available-choice factor6 = 16963914240 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, S026 S028 · calculationBefore Selection full ordered ORIGINAL available-choice factor7 := mul(16963914240, 25) After Selection full ordered ORIGINAL available-choice factor7 = 424097856000 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, S027 S029 · calculationBefore Selection full ordered ORIGINAL available-choice factor8 := mul(424097856000, 24) After Selection full ordered ORIGINAL available-choice factor8 = 10178348544000 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, S028 S030 · calculationBefore Selection full ordered ORIGINAL available-choice factor9 := mul(10178348544000, 23) After Selection full ordered ORIGINAL available-choice factor9 = 234102016512000 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, S029 S031 · calculationBefore Selection full ordered ORIGINAL available-choice factor10 := mul(234102016512000, 22) After Selection full ordered ORIGINAL available-choice factor10 = 5150244363264000 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, S030 S032 · calculationBefore Selection full ordered ORIGINAL available-choice factor11 := mul(5150244363264000, 21) After Selection full ordered ORIGINAL available-choice factor11 = 108155131628544000 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, S031 S033 · calculationBefore Selection full ordered ORIGINAL available-choice factor12 := mul(108155131628544000, 20) After Selection full ordered ORIGINAL available-choice factor12 = 2163102632570880000 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, S032 S034 · calculationBefore Selection full ordered ORIGINAL available-choice factor13 := mul(2163102632570880000, 19) After Selection full ordered ORIGINAL available-choice factor13 = 41098950018846720000 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, S033 S035 · calculationBefore Selection full ordered ORIGINAL available-choice factor14 := mul(41098950018846720000, 18) After Selection full ordered ORIGINAL available-choice factor14 = 739781100339240960000 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, S034 S036 · calculationBefore Selection full ordered ORIGINAL available-choice factor15 := mul(739781100339240960000, 17) After Selection full ordered ORIGINAL available-choice factor15 = 12576278705767096320000 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, S035 S037 · calculationBefore Selection full ordered ORIGINAL available-choice factor16 := mul(12576278705767096320000, 16) After Selection full ordered ORIGINAL available-choice factor16 = 201220459292273541120000 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, S036 S038 · calculationBefore Selection full ordered ORIGINAL available-choice factor17 := mul(201220459292273541120000, 15) After Selection full ordered ORIGINAL available-choice factor17 = 3018306889384103116800000 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, S037 S039 · calculationBefore Selection full ordered ORIGINAL available-choice factor18 := mul(3018306889384103116800000, 14) After Selection full ordered ORIGINAL available-choice factor18 = 42256296451377443635200000 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, S038 S040 · calculationBefore Selection full ordered ORIGINAL available-choice factor19 := mul(42256296451377443635200000, 13) After Selection full ordered ORIGINAL available-choice factor19 = 549331853867906767257600000 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, S039 S041 · calculationBefore Selection full ordered ORIGINAL available-choice factor20 := mul(549331853867906767257600000, 12) After Selection full ordered ORIGINAL available-choice factor20 = 6591982246414881207091200000 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, S040 S042 · calculationBefore Selection full ordered ORIGINAL available-choice factor21 := mul(6591982246414881207091200000, 11) After Selection full ordered ORIGINAL available-choice factor21 = 72511804710563693278003200000 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, S041 S043 · calculationBefore Selection full ordered ORIGINAL available-choice factor22 := mul(72511804710563693278003200000, 10) After Selection full ordered ORIGINAL available-choice factor22 = 725118047105636932780032000000 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, S042 S044 · calculationBefore Selection full ordered ORIGINAL available-choice factor23 := mul(725118047105636932780032000000, 9) After Selection full ordered ORIGINAL available-choice factor23 = 6526062423950732395020288000000 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, S043 S045 · calculationBefore Selection full ordered ORIGINAL available-choice factor24 := mul(6526062423950732395020288000000, 8) After Selection full ordered ORIGINAL available-choice factor24 = 52208499391605859160162304000000 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, S044 S046 · calculationBefore Selection full ordered ORIGINAL available-choice factor25 := mul(52208499391605859160162304000000, 7) After Selection full ordered ORIGINAL available-choice factor25 = 365459495741241014121136128000000 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, S045 S047 · calculationBefore Selection full ordered ORIGINAL available-choice factor26 := mul(365459495741241014121136128000000, 6) After Selection full ordered ORIGINAL available-choice factor26 = 2192756974447446084726816768000000 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, S046 S048 · calculationBefore Selection full ordered ORIGINAL available-choice factor27 := mul(2192756974447446084726816768000000, 5) After Selection full ordered ORIGINAL available-choice factor27 = 10963784872237230423634083840000000 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, S047 S049 · calculationBefore Selection full ordered ORIGINAL available-choice factor28 := mul(10963784872237230423634083840000000, 4) After Selection full ordered ORIGINAL available-choice factor28 = 43855139488948921694536335360000000 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, S048 S050 · calculationBefore Selection full ordered ORIGINAL available-choice factor29 := mul(43855139488948921694536335360000000, 3) After Selection full ordered ORIGINAL available-choice factor29 = 131565418466846765083609006080000000 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, S049 S051 · calculationBefore Selection full ordered ORIGINAL available-choice factor30 := mul(131565418466846765083609006080000000, 2) After Selection full ordered ORIGINAL available-choice factor30 = 263130836933693530167218012160000000 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, S050 S052 · calculationBefore Selection full ordered ORIGINAL available-choice factor31 := mul(263130836933693530167218012160000000, 1) After Selection full ordered ORIGINAL available-choice factor31 = 263130836933693530167218012160000000 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, S051 S053 · impliesBefore source roles = [Selection full ordered, 32, 32] After derived = [[32, 31, 30, 29, 28, 27, 26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1], 263130836933693530167218012160000000] 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, S052 S054 · calculationBefore Selection ORIGINAL completion COUNT ratio := div(20922789888000, 263130836933693530167218012160000000) After Selection ORIGINAL completion COUNT ratio = 1/12576278705767096320000 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, S053 S055 · impliesBefore source roles = [32, 32, 16, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]] After derived = [16, 16, 20922789888000, 263130836933693530167218012160000000, 1/12576278705767096320000] 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, S054 ## Reverse checks - S055 · 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. - S054 · premise_and_result_audit — Recompute Selection ORIGINAL completion COUNT ratio with exact rational arithmetic, not rounded decimals. - S053 · 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. - S052 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor31 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor30 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor29 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor28 with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor27 with exact rational arithmetic, not rounded decimals. - S047 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor26 with exact rational arithmetic, not rounded decimals. - S046 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor25 with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor24 with exact rational arithmetic, not rounded decimals. - S044 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor23 with exact rational arithmetic, not rounded decimals. - S043 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor22 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor21 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor20 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor19 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor18 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor17 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor16 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor15 with exact rational arithmetic, not rounded decimals. - S035 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor14 with exact rational arithmetic, not rounded decimals. - S034 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor13 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor12 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor11 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor10 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor9 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor8 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor7 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor6 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor5 with exact rational arithmetic, not rounded decimals. - S025 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor4 with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor3 with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor2 with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Selection full ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals. - S020 · 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. - S019 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor15 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor14 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor13 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor12 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor11 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor10 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor9 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor8 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor7 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor6 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor5 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor4 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor3 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor2 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Selection prefix remaining ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals. - 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 (16, 16, 20922789888000, 263130836933693530167218012160000000, 1/12576278705767096320000) 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: 29c987a6fe7e864ae9330dc96522934f6b1fd311f9f374009cb6911f4c745577 - candidate record SHA-256: 9f9ad2dd71a9686b73c95367526eeb4671ca936aec2a6b797181ed07e0468e94 - 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-0079.json)