# Question 35532: Boundary of the Basic Counting Principles

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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.

R0070-35532 · q-20260914-basic-counting-principle-0012



## Original question and givens

A process has 5 first-stage choices, but the number of second-stage options depends on the first choice and can be 5 or 6. May one fixed product be used unconditionally?


- G001 — A process has 5 first-stage choices, but the number of second-stage options depends on the first choice and can be 5 or 6. May one fixed product be used unconditionally?





## 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-counting-principle](https://ocme.evemisslab.com/en/mko/mko-counting-principle/)





## Step-by-step derivation

S001 · impliesBefore

source roles = [qf-basic-counting-principle, [branch_counts, conditional_stage_witness, mode, overlap_witness, stages]]

After

derived = [counting, null, null]

Conditions: none

Use THIS bank’s actual fields,operation/domain and types.BASE retains permissible extra metadata.Do not import LDL SAFE/set limits or a different bank’s deduplication rules.

Step identifiers and dependenciesRule: R70_SOURCE_bank_specific_interface · Uses: G001





S002 · impliesBefore

source roles = [5, 5, 6]

After

derived = [cannot multiply unconditionally]

Conditions: none

Two GIVEN conditional continuation counts differ,u≠v,so there is no common fixed b for an unconditional a·b total.A full total requires every first choice’s continuation count and summation.Unused stage metadata is not an extra premise.

Step identifiers and dependenciesRule: Counting_CONDITIONAL_different_continuations_no_fixed_product · Uses: G001, S001









## Reverse checks


- S002 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.
- S001 · premise_and_result_audit — Reverse-audit THIS question’s original operands,roles and necessary premises;reconstruct the next step,without reversing a one-way implication.Numerical/finite-table checking is not formal proof of an arbitrary theorem.Original Q,bilingual sources,MKO,AST,status and history remain preserved.





## Result

Cannot multiply unconditionally

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: 002d035de35b6cb42c4765937815e18ec2829b3907b8c9df89888d7f0ba7a8c3
- candidate record SHA-256: d6d966d183bfdcf20986cad21c7348cc9c4d5807bb396f4bd36262362caa65df
- semantic review receipt SHA-256: 9820ac6529139bd53d516c8bfdfdd41184b0cd52dad69fa433344010db37af80
- question-first witness SHA-256: c9c939762bbd4d3ea3403084c8e95541dc2c7e56eb82fc74eaba8c80a4517dab
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260914-basic-counting-principle-0012.json)