Question 6786: 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=8≤hi=11≤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=8≤hi=11≤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=8;hi=11;window factors=[9, 10, 11]

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

window through factor9 := mul(1, 9)

window through factor9 = 9

window through factor10 := mul(9, 10)

window through factor10 = 90

window through factor11 := mul(90, 11)

window through factor11 = 990

Reverse checks

• S005 · premise_and_result_audit — Recompute window through factor11 with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute window through factor10 with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute window through factor9 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

(990, [9, 10, 11], 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.