# Question 30878: Combinations Including or Excluding a Fixed Existing Label [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-0053) 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. R0060-30878 · q-20261011-advance-20261015-r06-permutation-combination-0053 ## 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=20≥1, k=10, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole). - 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=20≥1, k=10, fixed existing label 0. Give (all combinations, combinations containing 0, combinations excluding 0, sum of the two parts minus the whole). ## 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-one-distinguished-counts, density15_selection_one_distinguished_counts, finite_distinct_selection_protocol] After derived = [[total, included_label0, excluded_label0, partition_defect]] 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 all subsets ordered ORIGINAL available-choice factor0 := mul(1, 20) After Selection all subsets ordered ORIGINAL available-choice factor0 = 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 dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Selection all subsets ordered ORIGINAL available-choice factor1 := mul(20, 19) After Selection all subsets ordered ORIGINAL available-choice factor1 = 380 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 all subsets ordered ORIGINAL available-choice factor2 := mul(380, 18) After Selection all subsets ordered ORIGINAL available-choice factor2 = 6840 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 all subsets ordered ORIGINAL available-choice factor3 := mul(6840, 17) After Selection all subsets ordered ORIGINAL available-choice factor3 = 116280 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 all subsets ordered ORIGINAL available-choice factor4 := mul(116280, 16) After Selection all subsets ordered ORIGINAL available-choice factor4 = 1860480 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 all subsets ordered ORIGINAL available-choice factor5 := mul(1860480, 15) After Selection all subsets ordered ORIGINAL available-choice factor5 = 27907200 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 all subsets ordered ORIGINAL available-choice factor6 := mul(27907200, 14) After Selection all subsets ordered ORIGINAL available-choice factor6 = 390700800 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 all subsets ordered ORIGINAL available-choice factor7 := mul(390700800, 13) After Selection all subsets ordered ORIGINAL available-choice factor7 = 5079110400 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 all subsets ordered ORIGINAL available-choice factor8 := mul(5079110400, 12) After Selection all subsets ordered ORIGINAL available-choice factor8 = 60949324800 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 all subsets ordered ORIGINAL available-choice factor9 := mul(60949324800, 11) After Selection all subsets ordered ORIGINAL available-choice factor9 = 670442572800 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 all subsets ordered, 20, 10] After derived = [[20, 19, 18, 17, 16, 15, 14, 13, 12, 11], 670442572800] 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, S011 S013 · calculationBefore Selection all subsets within-subset ORIGINAL factorial factor2 := mul(1, 2) After Selection all subsets 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, S012 S014 · calculationBefore Selection all subsets within-subset ORIGINAL factorial factor3 := mul(2, 3) After Selection all subsets 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, S013 S015 · calculationBefore Selection all subsets within-subset ORIGINAL factorial factor4 := mul(6, 4) After Selection all subsets within-subset ORIGINAL factorial factor4 = 24 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 all subsets within-subset ORIGINAL factorial factor5 := mul(24, 5) After Selection all subsets within-subset ORIGINAL factorial factor5 = 120 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 all subsets within-subset ORIGINAL factorial factor6 := mul(120, 6) After Selection all subsets within-subset ORIGINAL factorial factor6 = 720 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 all subsets within-subset ORIGINAL factorial factor7 := mul(720, 7) After Selection all subsets within-subset ORIGINAL factorial factor7 = 5040 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 all subsets within-subset ORIGINAL factorial factor8 := mul(5040, 8) After Selection all subsets within-subset ORIGINAL factorial factor8 = 40320 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 · calculationBefore Selection all subsets within-subset ORIGINAL factorial factor9 := mul(40320, 9) After Selection all subsets within-subset ORIGINAL factorial factor9 = 362880 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, S019 S021 · calculationBefore Selection all subsets within-subset ORIGINAL factorial factor10 := mul(362880, 10) After Selection all subsets within-subset ORIGINAL factorial factor10 = 3628800 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 · impliesBefore source roles = [Selection all subsets within-subset, 10] After derived = [3628800] 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, S021 S023 · calculationBefore Selection all subsets ORIGINAL exact ordered over k factorial := div(670442572800, 3628800) After Selection all subsets ORIGINAL exact ordered over k factorial = 184756 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 all subsets reverse unordered times k factorial := mul(184756, 3628800) After Selection all subsets reverse unordered times k factorial = 670442572800 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 · impliesBefore source roles = [Selection all subsets, 20, 10, 670442572800, 3628800] After derived = [184756, 670442572800] 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, S024 S026 · calculationBefore Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 19) After Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 = 19 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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(19, 18) After Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 = 342 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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor2 := mul(342, 17) After Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor2 = 5814 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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor3 := mul(5814, 16) After Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor3 = 93024 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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor4 := mul(93024, 15) After Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor4 = 1395360 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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor5 := mul(1395360, 14) After Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor5 = 19535040 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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor6 := mul(19535040, 13) After Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor6 = 253955520 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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor7 := mul(253955520, 12) After Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor7 = 3047466240 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 subsets INCLUDING label0 ordered ORIGINAL available-choice factor8 := mul(3047466240, 11) After Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor8 = 33522128640 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 · impliesBefore source roles = [Selection subsets INCLUDING label0 ordered, 19, 9] After derived = [[19, 18, 17, 16, 15, 14, 13, 12, 11], 33522128640] 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, S034 S036 · calculationBefore Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor2 := mul(1, 2) After Selection subsets INCLUDING label0 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, S035 S037 · calculationBefore Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor3 := mul(2, 3) After Selection subsets INCLUDING label0 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, S036 S038 · calculationBefore Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor4 := mul(6, 4) After Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor4 = 24 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 subsets INCLUDING label0 within-subset ORIGINAL factorial factor5 := mul(24, 5) After Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor5 = 120 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 subsets INCLUDING label0 within-subset ORIGINAL factorial factor6 := mul(120, 6) After Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor6 = 720 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 subsets INCLUDING label0 within-subset ORIGINAL factorial factor7 := mul(720, 7) After Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor7 = 5040 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 subsets INCLUDING label0 within-subset ORIGINAL factorial factor8 := mul(5040, 8) After Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor8 = 40320 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 subsets INCLUDING label0 within-subset ORIGINAL factorial factor9 := mul(40320, 9) After Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor9 = 362880 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 · impliesBefore source roles = [Selection subsets INCLUDING label0 within-subset, 9] After derived = [362880] 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, S043 S045 · calculationBefore Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial := div(33522128640, 362880) After Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial = 92378 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 subsets INCLUDING label0 reverse unordered times k factorial := mul(92378, 362880) After Selection subsets INCLUDING label0 reverse unordered times k factorial = 33522128640 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 · impliesBefore source roles = [Selection subsets INCLUDING label0, 19, 9, 33522128640, 362880] After derived = [92378, 33522128640] 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, S046 S048 · calculationBefore Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 19) After Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 = 19 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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(19, 18) After Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 = 342 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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 := mul(342, 17) After Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 = 5814 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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor3 := mul(5814, 16) After Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor3 = 93024 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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor4 := mul(93024, 15) After Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor4 = 1395360 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 · calculationBefore Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor5 := mul(1395360, 14) After Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor5 = 19535040 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, S052 S054 · calculationBefore Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor6 := mul(19535040, 13) After Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor6 = 253955520 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 · calculationBefore Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor7 := mul(253955520, 12) After Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor7 = 3047466240 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, S054 S056 · calculationBefore Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor8 := mul(3047466240, 11) After Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor8 = 33522128640 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, S055 S057 · calculationBefore Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor9 := mul(33522128640, 10) After Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor9 = 335221286400 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, S056 S058 · impliesBefore source roles = [Selection subsets EXCLUDING label0 ordered, 19, 10] After derived = [[19, 18, 17, 16, 15, 14, 13, 12, 11, 10], 335221286400] 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, S057 S059 · calculationBefore Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor2 := mul(1, 2) After Selection subsets EXCLUDING label0 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, S058 S060 · calculationBefore Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor3 := mul(2, 3) After Selection subsets EXCLUDING label0 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, S059 S061 · calculationBefore Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor4 := mul(6, 4) After Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor4 = 24 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, S060 S062 · calculationBefore Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor5 := mul(24, 5) After Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor5 = 120 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, S061 S063 · calculationBefore Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor6 := mul(120, 6) After Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor6 = 720 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, S062 S064 · calculationBefore Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor7 := mul(720, 7) After Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor7 = 5040 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, S063 S065 · calculationBefore Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor8 := mul(5040, 8) After Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor8 = 40320 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, S064 S066 · calculationBefore Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor9 := mul(40320, 9) After Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor9 = 362880 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, S065 S067 · calculationBefore Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor10 := mul(362880, 10) After Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor10 = 3628800 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, S066 S068 · impliesBefore source roles = [Selection subsets EXCLUDING label0 within-subset, 10] After derived = [3628800] 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, S067 S069 · calculationBefore Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial := div(335221286400, 3628800) After Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial = 92378 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, S068 S070 · calculationBefore Selection subsets EXCLUDING label0 reverse unordered times k factorial := mul(92378, 3628800) After Selection subsets EXCLUDING label0 reverse unordered times k factorial = 335221286400 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, S069 S071 · impliesBefore source roles = [Selection subsets EXCLUDING label0, 19, 10, 335221286400, 3628800] After derived = [92378, 335221286400] 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, S070 S072 · calculationBefore Selection ORIGINAL included plus excluded := sum(92378, 92378) After Selection ORIGINAL included plus excluded = 184756 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, S071 S073 · calculationBefore Selection ORIGINAL partition sum minus whole := sub(184756, 184756) After Selection ORIGINAL partition sum minus whole = 0 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, S072 S074 · impliesBefore source roles = [20, 10] After derived = [184756, 92378, 92378, 0] Conditions: none n>=1 ensures the GIVEN existing label0.Containing it leaves k-1 choices from the other n-1 objects;excluding it chooses k from those n-1.The two classes are disjoint and exhaustive.At k0 inclusion count0,and at k=n exclusion count0 are legal. Step identifiers and dependenciesRule: Selection_FIXED_EXISTING_label0_disjoint_complete_subset_partition · Uses: G001, S073 ## Reverse checks - S074 · 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. - S073 · premise_and_result_audit — Recompute Selection ORIGINAL partition sum minus whole with exact rational arithmetic, not rounded decimals. - S072 · premise_and_result_audit — Recompute Selection ORIGINAL included plus excluded with exact rational arithmetic, not rounded decimals. - S071 · 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. - S070 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 reverse unordered times k factorial with exact rational arithmetic, not rounded decimals. - S069 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial with exact rational arithmetic, not rounded decimals. - S068 · 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. - S067 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor10 with exact rational arithmetic, not rounded decimals. - S066 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor9 with exact rational arithmetic, not rounded decimals. - S065 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor8 with exact rational arithmetic, not rounded decimals. - S064 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor7 with exact rational arithmetic, not rounded decimals. - S063 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor6 with exact rational arithmetic, not rounded decimals. - S062 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals. - S061 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals. - S060 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals. - S059 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 within-subset ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals. - S058 · 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. - S057 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor9 with exact rational arithmetic, not rounded decimals. - S056 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor8 with exact rational arithmetic, not rounded decimals. - S055 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor7 with exact rational arithmetic, not rounded decimals. - S054 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor6 with exact rational arithmetic, not rounded decimals. - S053 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor5 with exact rational arithmetic, not rounded decimals. - S052 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor4 with exact rational arithmetic, not rounded decimals. - S051 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor3 with exact rational arithmetic, not rounded decimals. - S050 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor2 with exact rational arithmetic, not rounded decimals. - S049 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals. - S048 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals. - S047 · 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. - S046 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 reverse unordered times k factorial with exact rational arithmetic, not rounded decimals. - S045 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial with exact rational arithmetic, not rounded decimals. - S044 · 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. - S043 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor9 with exact rational arithmetic, not rounded decimals. - S042 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor8 with exact rational arithmetic, not rounded decimals. - S041 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor7 with exact rational arithmetic, not rounded decimals. - S040 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor6 with exact rational arithmetic, not rounded decimals. - S039 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals. - S038 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals. - S037 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals. - S036 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 within-subset ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals. - S035 · 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. - S034 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor8 with exact rational arithmetic, not rounded decimals. - S033 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor7 with exact rational arithmetic, not rounded decimals. - S032 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor6 with exact rational arithmetic, not rounded decimals. - S031 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor5 with exact rational arithmetic, not rounded decimals. - S030 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor4 with exact rational arithmetic, not rounded decimals. - S029 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor3 with exact rational arithmetic, not rounded decimals. - S028 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor2 with exact rational arithmetic, not rounded decimals. - S027 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals. - S026 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals. - S025 · 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. - S024 · premise_and_result_audit — Recompute Selection all subsets reverse unordered times k factorial with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Selection all subsets ORIGINAL exact ordered over k factorial with exact rational arithmetic, not rounded decimals. - S022 · 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. - S021 · premise_and_result_audit — Recompute Selection all subsets within-subset ORIGINAL factorial factor10 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute Selection all subsets within-subset ORIGINAL factorial factor9 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute Selection all subsets within-subset ORIGINAL factorial factor8 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Selection all subsets within-subset ORIGINAL factorial factor7 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Selection all subsets within-subset ORIGINAL factorial factor6 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Selection all subsets within-subset ORIGINAL factorial factor5 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute Selection all subsets within-subset ORIGINAL factorial factor4 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Selection all subsets within-subset ORIGINAL factorial factor3 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Selection all subsets within-subset 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 all subsets ordered ORIGINAL available-choice factor9 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Selection all subsets ordered ORIGINAL available-choice factor8 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Selection all subsets ordered ORIGINAL available-choice factor7 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute Selection all subsets ordered ORIGINAL available-choice factor6 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Selection all subsets ordered ORIGINAL available-choice factor5 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Selection all subsets ordered ORIGINAL available-choice factor4 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Selection all subsets ordered ORIGINAL available-choice factor3 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Selection all subsets ordered ORIGINAL available-choice factor2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Selection all subsets ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Selection all subsets 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 (184756, 92378, 92378, 0) 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: 11a263caa4ed466e5d68682a20cd45e23c8f846fcc78b468cec2fbb2bd241104 - candidate record SHA-256: fc4ca684b4a3e074bde184660e9ed3ea426ea3de40af360fc8e9b39b01eb3468 - 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-0053.json)