# Question 30900: 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-0075) 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. R0060-30900 · q-20261011-advance-20261015-r06-permutation-combination-0075 ## 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=25≥1, k=2, 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=25≥1, k=2, 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, 25) After Selection all subsets ordered ORIGINAL available-choice factor0 = 25 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(25, 24) After Selection all subsets ordered ORIGINAL available-choice factor1 = 600 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 · impliesBefore source roles = [Selection all subsets ordered, 25, 2] After derived = [[25, 24], 600] 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, S003 S005 · 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, S004 S006 · impliesBefore source roles = [Selection all subsets 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 dependenciesRule: Selection_factorial_COMPLETE_integer_product_EMPTY0factorial1 · Uses: G001, S005 S007 · calculationBefore Selection all subsets ORIGINAL exact ordered over k factorial := div(600, 2) After Selection all subsets ORIGINAL exact ordered over k factorial = 300 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 reverse unordered times k factorial := mul(300, 2) After Selection all subsets reverse unordered times k factorial = 600 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 all subsets, 25, 2, 600, 2] After derived = [300, 600] 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, S008 S010 · calculationBefore Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 24) After Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 = 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, S009 S011 · impliesBefore source roles = [Selection subsets INCLUDING label0 ordered, 24, 1] After derived = [[24], 24] 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, S010 S012 · impliesBefore source roles = [Selection subsets INCLUDING label0 within-subset, 1] After derived = [1] 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 subsets INCLUDING label0 ORIGINAL exact ordered over k factorial := div(24, 1) After Selection subsets INCLUDING label0 ORIGINAL exact ordered over k factorial = 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, S012 S014 · calculationBefore Selection subsets INCLUDING label0 reverse unordered times k factorial := mul(24, 1) After Selection subsets INCLUDING label0 reverse unordered times k factorial = 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, S013 S015 · impliesBefore source roles = [Selection subsets INCLUDING label0, 24, 1, 24, 1] After derived = [24, 24] 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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 := mul(1, 24) After Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 = 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, S015 S017 · calculationBefore Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 := mul(24, 23) After Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 = 552 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 subsets EXCLUDING label0 ordered, 24, 2] After derived = [[24, 23], 552] 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, S017 S019 · 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, S018 S020 · impliesBefore source roles = [Selection subsets EXCLUDING label0 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 dependenciesRule: Selection_factorial_COMPLETE_integer_product_EMPTY0factorial1 · Uses: G001, S019 S021 · calculationBefore Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial := div(552, 2) After Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial = 276 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 · calculationBefore Selection subsets EXCLUDING label0 reverse unordered times k factorial := mul(276, 2) After Selection subsets EXCLUDING label0 reverse unordered times k factorial = 552 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, S021 S023 · impliesBefore source roles = [Selection subsets EXCLUDING label0, 24, 2, 552, 2] After derived = [276, 552] 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, S022 S024 · calculationBefore Selection ORIGINAL included plus excluded := sum(24, 276) After Selection ORIGINAL included plus excluded = 300 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 · calculationBefore Selection ORIGINAL partition sum minus whole := sub(300, 300) 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, S024 S026 · impliesBefore source roles = [25, 2] After derived = [300, 24, 276, 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, S025 ## Reverse checks - S026 · 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. - S025 · premise_and_result_audit — Recompute Selection ORIGINAL partition sum minus whole with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Selection ORIGINAL included plus excluded with exact rational arithmetic, not rounded decimals. - S023 · 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. - S022 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 reverse unordered times k factorial with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ORIGINAL exact ordered over k factorial 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 subsets EXCLUDING label0 within-subset ORIGINAL factorial factor2 with exact rational arithmetic, not rounded decimals. - 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 subsets EXCLUDING label0 ordered ORIGINAL available-choice factor1 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute Selection subsets EXCLUDING label0 ordered ORIGINAL available-choice factor0 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 subsets INCLUDING label0 reverse unordered times k factorial with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 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 — 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. - S010 · premise_and_result_audit — Recompute Selection subsets INCLUDING label0 ordered ORIGINAL available-choice factor0 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 all subsets reverse unordered times k factorial with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute Selection all subsets ORIGINAL exact ordered over k factorial with exact rational arithmetic, not rounded decimals. - S006 · 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. - S005 · premise_and_result_audit — Recompute Selection all subsets within-subset ORIGINAL factorial factor2 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 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 (300, 24, 276, 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: adf80306eb329e2e3be4235bb1588fcc76a665e3935ad423c76c40577cc1c2eb - candidate record SHA-256: a2c07f74c2880b5dfa66435929b6d3557d4f50fc7c79f787626ed09f6e50083f - 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-0075.json)