Question 6778: Factorial Window Product and Empty-Window Eligibility

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

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

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]

hi!/lo! is the product of exactly these factors

window through factor1 := mul(1, 1)

window through factor1 = 1

window through factor2 := mul(1, 2)

window through factor2 = 2

window through factor3 := mul(2, 3)

window through factor3 = 6

window through factor4 := mul(6, 4)

window through factor4 = 24

window through factor5 := mul(24, 5)

window through factor5 = 120

window through factor6 := mul(120, 6)

window through factor6 = 720

window through factor7 := mul(720, 7)

window through factor7 = 5040

window through factor8 := mul(5040, 8)

window through factor8 = 40320

window through factor9 := mul(40320, 9)

window through factor9 = 362880

window through factor10 := mul(362880, 10)

window through factor10 = 3628800

window through factor11 := mul(3628800, 11)

window through factor11 = 39916800

window through factor12 := mul(39916800, 12)

window through factor12 = 479001600

window through factor13 := mul(479001600, 13)

window through factor13 = 6227020800

window through factor14 := mul(6227020800, 14)

window through factor14 = 87178291200

window through factor15 := mul(87178291200, 15)

window through factor15 = 1307674368000

window through factor16 := mul(1307674368000, 16)

window through factor16 = 20922789888000

window through factor17 := mul(20922789888000, 17)

window through factor17 = 355687428096000

window through factor18 := mul(355687428096000, 18)

window through factor18 = 6402373705728000

window through factor19 := mul(6402373705728000, 19)

window through factor19 = 121645100408832000

window through factor20 := mul(121645100408832000, 20)

window through factor20 = 2432902008176640000

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.