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=15, k=3, r=1; 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=15, k=3, r=1; 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).
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(15, 1)
After
Selection prefix ORIGINAL n minus used r = 14
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(3, 1)
After
Selection prefix ORIGINAL k minus filled r = 2
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, 14)
After
Selection prefix remaining ordered ORIGINAL available-choice factor0 = 14
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(14, 13)
After
Selection prefix remaining ordered ORIGINAL available-choice factor1 = 182
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 · implies
Before
source roles = [Selection prefix remaining ordered, 14, 2]
After
derived = [[14, 13], 182]
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, S005
S007 · calculation
Before
Selection full ordered ORIGINAL available-choice factor0 := mul(1, 15)
After
Selection full ordered ORIGINAL available-choice factor0 = 15
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 full ordered ORIGINAL available-choice factor1 := mul(15, 14)
After
Selection full ordered ORIGINAL available-choice factor1 = 210
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 full ordered ORIGINAL available-choice factor2 := mul(210, 13)
After
Selection full ordered ORIGINAL available-choice factor2 = 2730
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 · implies
Before
source roles = [Selection full ordered, 15, 3]
After
derived = [[15, 14, 13], 2730]
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, S009
S011 · calculation
Before
Selection ORIGINAL completion COUNT ratio := div(182, 2730)
After
Selection ORIGINAL completion COUNT ratio = 1/15
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 · implies
Before
source roles = [15, 3, 1, [0]]
After
derived = [14, 2, 182, 2730, 1/15]
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, S011