# Question 6766: Omitting One Existing Factor and Multiplying Back [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-0091) 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. R0012-06766 · q-20261010-advance-20261014-r05-factorial-0091 ## Original question and givens N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=17, k=16∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!). - G001 — N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=17, k=16∈[1,n]. Give (n!, product with k omitted once n!/k, k times that product minus n!). ## 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 actual factorial:start at F(0) After F(0)=1 Conditions: none The product of zero factors is1,not0;this is the successor recurrence's base. Step identifiers and dependenciesRule: factorial_empty_product_base · Uses: G001, S001 S003 · calculationBefore actual factorial through factor1 := mul(1, 1) After actual factorial through factor1 = 1 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore actual factorial through factor2 := mul(1, 2) After actual factorial through factor2 = 2 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore actual factorial through factor3 := mul(2, 3) After actual factorial through factor3 = 6 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore actual factorial through factor4 := mul(6, 4) After actual factorial through factor4 = 24 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore actual factorial through factor5 := mul(24, 5) After actual factorial through factor5 = 120 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore actual factorial through factor6 := mul(120, 6) After actual factorial through factor6 = 720 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore actual factorial through factor7 := mul(720, 7) After actual factorial through factor7 = 5040 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore actual factorial through factor8 := mul(5040, 8) After actual factorial through factor8 = 40320 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore actual factorial through factor9 := mul(40320, 9) After actual factorial through factor9 = 362880 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore actual factorial through factor10 := mul(362880, 10) After actual factorial through factor10 = 3628800 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore actual factorial through factor11 := mul(3628800, 11) After actual factorial through factor11 = 39916800 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore actual factorial through factor12 := mul(39916800, 12) After actual factorial through factor12 = 479001600 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore actual factorial through factor13 := mul(479001600, 13) After actual factorial through factor13 = 6227020800 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore actual factorial through factor14 := mul(6227020800, 14) After actual factorial through factor14 = 87178291200 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · calculationBefore actual factorial through factor15 := mul(87178291200, 15) After actual factorial through factor15 = 1307674368000 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore actual factorial through factor16 := mul(1307674368000, 16) After actual factorial through factor16 = 20922789888000 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore actual factorial through factor17 := mul(20922789888000, 17) After actual factorial through factor17 = 355687428096000 Conditions: none Multiply the preceding factorial by the new positive integer factor,retaining every exact intermediate. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · calculationBefore single-factor-omitted product := div(355687428096000, 16) After single-factor-omitted product = 22230464256000 Conditions: none The nonzero k occurs exactly once in the1,...,n factor list;removing that factor gives n!/k. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · calculationBefore restore omitted factor := mul(16, 22230464256000) After restore omitted factor = 355687428096000 Conditions: none Multiply the same k back to audit the complete product. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore restored product minus factorial := sub(355687428096000, 355687428096000) After restored product minus factorial = 0 Conditions: none Restored product minus the true factorial must be0. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore removed factor k=16;full n!=355687428096000 After omitted product=22230464256000;restoration residual=0 Conditions: none Omit the designated factor once,not every factor divisible by k;check by exact positive-integer division and reverse multiplication. Step identifiers and dependenciesRule: single_factor_removal_reverse_audit · Uses: G001, S022 ## Reverse checks - S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S022 · premise_and_result_audit — Recompute restored product minus factorial with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute restore omitted factor with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Recompute single-factor-omitted product with exact rational arithmetic, not rounded decimals. - S019 · premise_and_result_audit — Recompute actual factorial through factor17 with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute actual factorial through factor16 with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute actual factorial through factor15 with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Recompute actual factorial through factor14 with exact rational arithmetic, not rounded decimals. - S015 · premise_and_result_audit — Recompute actual factorial through factor13 with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute actual factorial through factor12 with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute actual factorial through factor11 with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Recompute actual factorial through factor10 with exact rational arithmetic, not rounded decimals. - S011 · premise_and_result_audit — Recompute actual factorial through factor9 with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute actual factorial through factor8 with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute actual factorial through factor7 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute actual factorial through factor6 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute actual factorial through factor5 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute actual factorial through factor4 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute actual factorial through factor3 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute actual factorial through factor2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute actual factorial 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 (355687428096000, 22230464256000, 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: d1135b6ff8a5e1c565b894b01813c07a71ad24be7a7a043c5c8f9ce38dd3b349 - candidate record SHA-256: bdd36727e8f73b8cb8ff964ed411fa775750aa2f4530e487fc7d2b2e0222b727 - 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-0091.json)