# Question 5328: Disjoint Tagged Choices and Prefix Counts

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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.

R0009-05328 · q-20261008-advance-20261012-r02-counting-principle-0003



## Original question and givens

Mutually exclusive tagged categories have ordered sizes=[1]. Give (total choices, each prefix’s total), retaining an empty list.


- G001 — Mutually exclusive tagged categories have ordered sizes=[1]. Give (total choices, each prefix’s total), retaining an empty list.





## 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

declared finite tagged choices or full-product protocol

After

combinatorial counts,not probabilities or independence

Conditions: none

Addition uses disjoint tags and multiplication uses the declared full cross-stage protocol;neither is a probability-independence assumption.

Step identifiers and dependenciesRule: finite_counting_protocol_scope · Uses: G001





S002 · impliesBefore

ordered sizes=[1]

After

initial accumulator=0

Conditions: none

The empty disjoint sum is0;the empty full product is1,representing one empty choice sequence.

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





S003 · calculationBefore

prefix count1 := sum(0, 1)

After

prefix count1 = 1

Conditions: none

Multiply in a stage of the declared full product,or add a disjoint tagged class,retaining every prefix.

Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002





S004 · impliesBefore

prefix through position0;new size=1

After

prefix total=1

Conditions: none

A full product stays0 after a zero factor;the disjoint sum continues adding in order.Do not interchange these protocols.

Step identifiers and dependenciesRule: counting_prefix_update · Uses: G001, S003









## Reverse checks


- S004 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
- S003 · premise_and_result_audit — Recompute prefix count1 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

(1, [1])

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: 451642d64c668cfa5998722b1f1835ac3a6aad08cc46fca2f8750d25890fa245
- candidate record SHA-256: 7b41f1f6bc08a106464119345ec630a71b76b8a01710658eea4efb0c52a01834
- semantic review receipt SHA-256: 71e7db519df31f3a164533e472e293cb64f001b475a8d002b4a5c2740a237b56
- question-first witness SHA-256: 5a43f4f81b6256281f8d139a4010bbde127be803c528e5878857269c8f854462
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261012-r02-counting-principle-0003.json)