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Question 30905: Ordered/Unordered Counts, Factorial, and Empty Selection

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

R0060-30905 · q-20261011-advance-20261015-r06-permutation-combination-0005

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=5, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

  • 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=5, k=2, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).

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-order-factor-triple, density15_selection_order_factor_triple, finite_distinct_selection_protocol]

After

derived = [[ordered_count, unordered_count, ordering_factor, empty_selection]]

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 original full ordered ORIGINAL available-choice factor0 := mul(1, 5)

After

Selection original full ordered ORIGINAL available-choice factor0 = 5

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 original full ordered ORIGINAL available-choice factor1 := mul(5, 4)

After

Selection original full ordered ORIGINAL available-choice factor1 = 20

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 · implies

Before

source roles = [Selection original full ordered, 5, 2]

After

derived = [[5, 4], 20]

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, S003

S005 · calculation

Before

Selection original unordered ordered ORIGINAL available-choice factor0 := mul(1, 5)

After

Selection original unordered ordered ORIGINAL available-choice factor0 = 5

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 original unordered ordered ORIGINAL available-choice factor1 := mul(5, 4)

After

Selection original unordered ordered ORIGINAL available-choice factor1 = 20

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 · implies

Before

source roles = [Selection original unordered ordered, 5, 2]

After

derived = [[5, 4], 20]

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, S006

S008 · calculation

Before

Selection original unordered within-subset ORIGINAL factorial factor2 := mul(1, 2)

After

Selection original unordered within-subset ORIGINAL factorial factor2 = 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, S007

S009 · implies

Before

source roles = [Selection original unordered within-subset, 2]

After

derived = [2]

Conditions: none

Multiply the COMPLETE2..n factors;at n0/1 the empty product is1.Use exact BigInt,not floats or a SAFE cap on derived counts.

Step identifiers and dependencies

Rule: Selection_factorial_COMPLETE_integer_product_EMPTY0factorial1 · Uses: G001, S008

S010 · calculation

Before

Selection original unordered ORIGINAL exact ordered over k factorial := div(20, 2)

After

Selection original unordered ORIGINAL exact ordered over k factorial = 10

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 original unordered reverse unordered times k factorial := mul(10, 2)

After

Selection original unordered reverse unordered times k factorial = 20

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 = [Selection original unordered, 5, 2, 20, 2]

After

derived = [10, 20]

Conditions: none

EACH k-subset has exactly k! orderings,so P/k! is the exact integer C.Multiply back to verify P.An empty subset and empty ordering each count once;no probability model is asserted.

Step identifiers and dependencies

Rule: Selection_UNORDERED_each_subset_has_kfactorial_orderings · Uses: G001, S011

S013 · calculation

Before

Selection original ordering factor ORIGINAL factorial factor2 := mul(1, 2)

After

Selection original ordering factor ORIGINAL factorial factor2 = 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, S012

S014 · implies

Before

source roles = [Selection original ordering factor, 2]

After

derived = [2]

Conditions: none

Multiply the COMPLETE2..n factors;at n0/1 the empty product is1.Use exact BigInt,not floats or a SAFE cap on derived counts.

Step identifiers and dependencies

Rule: Selection_factorial_COMPLETE_integer_product_EMPTY0factorial1 · Uses: G001, S013

Reverse checks

  • S014 · 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.
  • S013 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
  • S012 · 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.
  • S011 · premise_and_result_audit — Recompute Selection original unordered reverse unordered times k factorial with exact rational arithmetic, not rounded decimals.
  • S010 · premise_and_result_audit — Recompute Selection original unordered ORIGINAL exact ordered over k factorial with exact rational arithmetic, not rounded decimals.
  • S009 · 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.
  • S008 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals.
  • 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 unordered ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals.
  • 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 original full ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor0 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

(20, 10, 2, No)

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: 1fba982f4272dd37c4b9292e28264ab3259e4ac678ef5242baf5019b3df1d5f0
  • candidate record SHA-256: 4201106ac593d20f22500df2c18a7c641b72eb008a414c0a1b74c14cf1619c7d
  • semantic review receipt SHA-256: d6b5a409bf933281a46617cd715d8016fd5ca3d6fb76f77d179ac2ecd9d608bf
  • question-first witness SHA-256: f80ecf598c6893a580af018e1a82849b33ee6887efb48a6a1d69569aa17f12e4
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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