# Question 6778: Factorial Window Product and Empty-Window Eligibility [Return to catalogue page 34](https://ocme.evemisslab.com/en/questions/page/34/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261010-advance-20261014-r05-factorial-foundations/#q-20261010-advance-20261014-r05-factorial-0028) 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. R0012-06778 · q-20261010-advance-20261014-r05-factorial-0028 ## Original question and givens N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=0≤hi=20≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?). - G001 — N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=0≤hi=20≤64. Give (hi!/lo!, factor indices lo+1 through hi, is the window empty?). ## 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-factorial](https://ocme.evemisslab.com/en/mko/mko-factorial/) ## Step-by-step derivation S001 · impliesBefore N includes0;F(0)=1;F(j)=j·F(j−1) After exact integer products,not floating values or a permutation proof Conditions: none Use empty product1 and integer factor multiplication,retaining large values exactly.This finite recurrence audit is not a general proof of permutation counting. Step identifiers and dependenciesRule: exact_factorial_protocol · Uses: G001 S002 · impliesBefore lo=0;hi=20;window factors=[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] After hi!/lo! is the product of exactly these factors Conditions: none The positive integer lo! permits cancellation of the common prefix;when lo=hi,no factors remain and the quotient is1. Step identifiers and dependenciesRule: factorial_positive_prefix_cancellation · Uses: G001, S001 S003 · calculationBefore window through factor1 := mul(1, 1) After window through factor1 = 1 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore window through factor2 := mul(1, 2) After window through factor2 = 2 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore window through factor3 := mul(2, 3) After window through factor3 = 6 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore window through factor4 := mul(6, 4) After window through factor4 = 24 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore window through factor5 := mul(24, 5) After window through factor5 = 120 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore window through factor6 := mul(120, 6) After window through factor6 = 720 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore window through factor7 := mul(720, 7) After window through factor7 = 5040 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore window through factor8 := mul(5040, 8) After window through factor8 = 40320 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore window through factor9 := mul(40320, 9) After window through factor9 = 362880 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore window through factor10 := mul(362880, 10) After window through factor10 = 3628800 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore window through factor11 := mul(3628800, 11) After window through factor11 = 39916800 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore window through factor12 := mul(39916800, 12) After window through factor12 = 479001600 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore window through factor13 := mul(479001600, 13) After window through factor13 = 6227020800 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore window through factor14 := mul(6227020800, 14) After window through factor14 = 87178291200 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore window through factor15 := mul(87178291200, 15) After window through factor15 = 1307674368000 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore window through factor16 := mul(1307674368000, 16) After window through factor16 = 20922789888000 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore window through factor17 := mul(20922789888000, 17) After window through factor17 = 355687428096000 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore window through factor18 := mul(355687428096000, 18) After window through factor18 = 6402373705728000 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore window through factor19 := mul(6402373705728000, 19) After window through factor19 = 121645100408832000 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore window through factor20 := mul(121645100408832000, 20) After window through factor20 = 2432902008176640000 Conditions: none Multiply only factors from lo+1 through hi,without outside factors. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 ## Reverse checks - S022 · premise_and_result_audit — Recompute window through factor20 with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute window through factor19 with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute window through factor18 with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute window through factor17 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute window through factor16 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute window through factor15 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute window through factor14 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute window through factor13 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute window through factor12 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute window through factor11 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute window through factor10 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute window through factor9 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute window through factor8 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute window through factor7 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute window through factor6 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute window through factor5 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute window through factor4 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute window through factor3 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute window through factor2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute window through factor1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result (2432902008176640000, [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], 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: 4c2c04075c652d76d53b3d8c66eb0e660cc0a01ee84a47bb2cf5ea79fd253743 - candidate record SHA-256: 740c0c7d516a395d4e773bddbd310c500e994b7d4dd6ac7876d1ef245eeec373 - semantic review receipt SHA-256: 90dbc494b2d34ef51f412dcff63cfcc1a2215d038bff71f38f5dded4c64faf00 - question-first witness SHA-256: b72a3b8bcf10f584396e0dd5198c657b0934a71864e5a3fa6321d7a613aef9f9 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261010-advance-20261014-r05-factorial-0028.json)