Question 6752: Omitting One Existing Factor and Multiplying Back

Original question and givens

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. n=19, k=19∈[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=19, k=19∈[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

Step-by-step derivation

N includes0;F(0)=1;F(j)=j·F(j−1)

exact integer products,not floating values or a permutation proof

actual factorial:start at F(0)

F(0)=1

actual factorial through factor1 := mul(1, 1)

actual factorial through factor1 = 1

actual factorial through factor2 := mul(1, 2)

actual factorial through factor2 = 2

actual factorial through factor3 := mul(2, 3)

actual factorial through factor3 = 6

actual factorial through factor4 := mul(6, 4)

actual factorial through factor4 = 24

actual factorial through factor5 := mul(24, 5)

actual factorial through factor5 = 120

actual factorial through factor6 := mul(120, 6)

actual factorial through factor6 = 720

actual factorial through factor7 := mul(720, 7)

actual factorial through factor7 = 5040

actual factorial through factor8 := mul(5040, 8)

actual factorial through factor8 = 40320

actual factorial through factor9 := mul(40320, 9)

actual factorial through factor9 = 362880

actual factorial through factor10 := mul(362880, 10)

actual factorial through factor10 = 3628800

actual factorial through factor11 := mul(3628800, 11)

actual factorial through factor11 = 39916800

actual factorial through factor12 := mul(39916800, 12)

actual factorial through factor12 = 479001600

actual factorial through factor13 := mul(479001600, 13)

actual factorial through factor13 = 6227020800

actual factorial through factor14 := mul(6227020800, 14)

actual factorial through factor14 = 87178291200

actual factorial through factor15 := mul(87178291200, 15)

actual factorial through factor15 = 1307674368000

actual factorial through factor16 := mul(1307674368000, 16)

actual factorial through factor16 = 20922789888000

actual factorial through factor17 := mul(20922789888000, 17)

actual factorial through factor17 = 355687428096000

actual factorial through factor18 := mul(355687428096000, 18)

actual factorial through factor18 = 6402373705728000

actual factorial through factor19 := mul(6402373705728000, 19)

actual factorial through factor19 = 121645100408832000

single-factor-omitted product := div(121645100408832000, 19)

single-factor-omitted product = 6402373705728000

restore omitted factor := mul(19, 6402373705728000)

restore omitted factor = 121645100408832000

restored product minus factorial := sub(121645100408832000, 121645100408832000)

restored product minus factorial = 0

removed factor k=19;full n!=121645100408832000

omitted product=6402373705728000;restoration residual=0

Reverse checks

• S025 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S024 · premise_and_result_audit — Recompute restored product minus factorial with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute restore omitted factor with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute single-factor-omitted product with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute actual factorial through factor19 with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute actual factorial through factor18 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

(121645100408832000, 6402373705728000, 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.