# Question 30908: Ordered/Unordered Counts, Factorial, and Empty Selection [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-0008) 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. R0060-30908 · q-20261011-advance-20261015-r06-permutation-combination-0008 ## 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=8, k=3, 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=8, k=3, 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. - [mko-permutation-combination](https://ocme.evemisslab.com/en/mko/mko-permutation-combination/) ## Step-by-step derivation S001 · impliesBefore 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 dependenciesRule: R60_EXACT_perbank_operation_domain_FRAME_and_output_roles · Uses: G001 S002 · calculationBefore Selection original full ordered ORIGINAL available-choice factor0 := mul(1, 8) After Selection original full ordered ORIGINAL available-choice factor0 = 8 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 original full ordered ORIGINAL available-choice factor1 := mul(8, 7) After Selection original full ordered ORIGINAL available-choice factor1 = 56 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 original full ordered ORIGINAL available-choice factor2 := mul(56, 6) After Selection original full ordered ORIGINAL available-choice factor2 = 336 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 · impliesBefore source roles = [Selection original full ordered, 8, 3] After derived = [[8, 7, 6], 336] Conditions: none At position j there are n-j UNUSED distinct labeled objects.Multiply ALL k available-choice counts;at k0 there is exactly ONE empty ordered tuple.This is not replacement,circular or indistinguishable selection. Step identifiers and dependenciesRule: Selection_ORDERED_distinct_without_replacement_available_choices · Uses: G001, S004 S006 · calculationBefore Selection original unordered ordered ORIGINAL available-choice factor0 := mul(1, 8) After Selection original unordered ordered ORIGINAL available-choice factor0 = 8 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 original unordered ordered ORIGINAL available-choice factor1 := mul(8, 7) After Selection original unordered ordered ORIGINAL available-choice factor1 = 56 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 original unordered ordered ORIGINAL available-choice factor2 := mul(56, 6) After Selection original unordered ordered ORIGINAL available-choice factor2 = 336 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 · impliesBefore source roles = [Selection original unordered ordered, 8, 3] After derived = [[8, 7, 6], 336] 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, S008 S010 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Selection original unordered within-subset ORIGINAL factorial factor3 := mul(2, 3) After Selection original unordered within-subset ORIGINAL factorial factor3 = 6 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 · impliesBefore source roles = [Selection original unordered within-subset, 3] After derived = [6] 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 dependenciesRule: Selection_factorial_COMPLETE_integer_product_EMPTY0factorial1 · Uses: G001, S011 S013 · calculationBefore Selection original unordered ORIGINAL exact ordered over k factorial := div(336, 6) After Selection original unordered ORIGINAL exact ordered over k factorial = 56 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 original unordered reverse unordered times k factorial := mul(56, 6) After Selection original unordered reverse unordered times k factorial = 336 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 · impliesBefore source roles = [Selection original unordered, 8, 3, 336, 6] After derived = [56, 336] 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 dependenciesRule: Selection_UNORDERED_each_subset_has_kfactorial_orderings · Uses: G001, S014 S016 · calculationBefore 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore Selection original ordering factor ORIGINAL factorial factor3 := mul(2, 3) After Selection original ordering factor ORIGINAL factorial factor3 = 6 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 · impliesBefore source roles = [Selection original ordering factor, 3] After derived = [6] 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 dependenciesRule: Selection_factorial_COMPLETE_integer_product_EMPTY0factorial1 · Uses: G001, S017 ## Reverse checks - S018 · 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. - S017 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Selection original ordering factor ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals. - S015 · 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. - S014 · premise_and_result_audit — Recompute Selection original unordered reverse unordered times k factorial with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Selection original unordered ORIGINAL exact ordered over k factorial 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 within-subset ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Selection original unordered within-subset ORIGINAL factorial factor2 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 ordered ORIGINAL available-choice factor2 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate. - S004 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor2 with exact rational arithmetic, not rounded decimals. - 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 (336, 56, 6, 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: 228955c615959b2648744bfdc3636ec0af83dce8d6cc077051d703d36aae2ee3 - candidate record SHA-256: 62c8614b404b2a1308a23da78bd6f435aa8aeabaff83febaa523fd8781b227d9 - 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-0008.json)