# Basic counting principles

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

The addition and multiplication principles combine finite counts across disjoint branches or successive choices.

mko-counting-principle · concept · v0.10.0



## Mathematical statement

Counts of mutually exclusive choice classes are added. Counts of combinations across successive choice stages are multiplied when the options and conditions at every stage are explicit.





## Representative formula (exact TeX)

N(A)





## Assumptions


- The choice sets or choice sequences being counted are finite.
- For the addition principle, the classes are disjoint or their overlap is corrected separately.
- For the multiplication principle, every stage and its number of available options are explicitly described relative to prior choices.





## Symbol scope


- N — finite cardinality function
- A — finite choice collection





## Intuition, definition, and misconceptions

If a task can be completed in one of m mutually exclusive ways or one of n mutually exclusive ways, the total is m+n. Disjointness matters because otherwise the same outcome may be counted twice.

If a process has m first-stage choices and each has n subsequent choices, there are m×n ordered combinations. If the number of later options depends on the first choice, the branches must be counted separately.

N(A) denotes only the number of elements in a finite set A. Complete formulas and formal proofs for the addition and multiplication principles are handled separately by later compiler and Evidence work.

A common mistake is to add overlapping cases directly or multiply unconditionally when later option counts depend on earlier choices.





## Adopted derivation guide

[Basic counting principles: how many three-digit even numbers have distinct digits](https://ocme.evemisslab.com/en/derivations/derivation-mko-counting-principle-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Natural number](https://ocme.evemisslab.com/en/mko/mko-natural-number/) — Counts of finite objects are represented by natural numbers.
- [Arithmetic operations](https://ocme.evemisslab.com/en/mko/mko-arithmetic-operations/) — Basic counting principles use addition and multiplication to combine local counts.





## Verification boundary

This object does not yet bind every condition of the addition and multiplication principles to formal Evidence; the representative expression N(A) is not a complete proof.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.





## Next obligation

Align the principles with Fintype.card_sum and Fintype.card_prod, making the no-overlap condition explicit for addition.





## Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/counting-principle.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-counting-principle/index.md)