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Question 30854: Completions after Fixing an Initial Ordered Prefix

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

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.

Step-by-step derivation

S001 · implies

Before

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 dependencies

Rule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001

S002 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S010

S012 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S011

S013 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S012

S014 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S013

S015 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S014

S016 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S015

S017 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S016

S018 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S017

S019 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S018

S020 · implies

Before

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 dependencies

Rule: Selection_ORDERED_distinct_without_replacement_available_choices · Uses: G001, S019

S021 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S020

S022 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S021

S023 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S022

S024 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S023

S025 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S024

S026 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S025

S027 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S026

S028 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S027

S029 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S028

S030 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S029

S031 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S030

S032 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S031

S033 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S032

S034 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S033

S035 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S034

S036 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S035

S037 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S036

S038 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S037

S039 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S038

S040 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S039

S041 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S040

S042 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S041

S043 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S042

S044 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S043

S045 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S044

S046 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S045

S047 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S046

S048 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S047

S049 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S048

S050 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S049

S051 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S050

S052 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S051

S053 · implies

Before

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 dependencies

Rule: Selection_ORDERED_distinct_without_replacement_available_choices · Uses: G001, S052

S054 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S053

S055 · implies

Before

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 dependencies

Rule: 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