# Question 6792: 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-0042)

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.

R0012-06792 · q-20261010-advance-20261014-r05-factorial-0042



## Original question and givens

N begins at 0, F(0)=0!=1, and factorials use exact integer arithmetic, not floating point. 0≤lo=14≤hi=17≤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=14≤hi=17≤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=14;hi=17;window factors=[15, 16, 17]

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 factor15 := mul(1, 15)

After

window through factor15 = 15

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 factor16 := mul(15, 16)

After

window through factor16 = 240

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 factor17 := mul(240, 17)

After

window through factor17 = 4080

Conditions: none

Multiply only factors from lo+1 through hi,without outside factors.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004









## Reverse checks


- S005 · premise_and_result_audit — Recompute window through factor17 with exact rational arithmetic, not rounded decimals.
- S004 · premise_and_result_audit — Recompute window through factor16 with exact rational arithmetic, not rounded decimals.
- S003 · premise_and_result_audit — Recompute window through factor15 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

(4080, [15, 16, 17], 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: 8bedb43c096b0f111cca39d2d4a8b880aa3b92d6a9cafd7ca82aeee1e879b963
- candidate record SHA-256: 37b79361204322192c76586738a62eb3d20faf3a1acc6dbb9ced36bddf4f86ed
- 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-0042.json)