# Question 3019: Exact Unordered Selection Count

[Return to catalogue page 16](https://ocme.evemisslab.com/en/questions/page/16/)

[← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20260930-r06-permutation-combination-foundations/#q-20260930-r06-combination-exact-count-0019)

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.

R0005-03019 · q-20260930-r06-combination-exact-count-0019



## Original question and givens

Select k=0 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,0).


- G001 — Select k=0 objects without order, replacement, or repetition from n=8 distinct objects. The empty selection counts once. Find C(8,0).





## 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-permutation-combination](https://ocme.evemisslab.com/en/mko/mko-permutation-combination/)





## Step-by-step derivation

S001 · impliesBefore

n=8;k=0;distinct objects,without replacement

After

unordered k-subsets of an n-element set

Conditions: none

Each object is selected at most once and order is ignored;the ordered-selection count is not the requested answer.

Step identifiers and dependenciesRule: unordered_selection_model · Uses: G001





S002 · impliesBefore

ordered selections=1;orders per subset=1

After

C(8,0)=ordered selections/k!

Conditions: none

Every k-subset has exactly k! orderings,and different subsets have disjoint orderings;divide by this common positive factor.

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





S003 · calculationBefore

unordered subset count := div(1, 1)

After

unordered subset count = 1

Conditions: none

Exact division gives an integer;when k=0 both empty products are1 and there is exactly one empty selection.

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





S004 · impliesBefore

C(8,0)=n!/(k!·(n−k)!)

After

C(8,0)=40320/(1·40320)=1

Conditions: none

Rewrite the falling product as n!/(n−k)! and audit the binomial formula,including boundary cases.

Step identifiers and dependenciesRule: binomial_count_formula · 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 unordered subset count 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

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: 71e2bd0b54e9794b2095c675c686e969fb50382addb494567cddee9ad15a57e8
- candidate record SHA-256: a3a6f8b6eae62e88d5b25a683f5f7e16df01373db181f638b6eec2ed2a9376db
- semantic review receipt SHA-256: 5d74de11610ebc6037a4fc9946d77154d4ed18389f8f0feeb823148b6320c4c2
- question-first witness SHA-256: 1fba4ab0f13c2cfcc4d9fdebe492c442d43dd45700394a443daa28f333355ed9
- Lean: NOT_RUN · human review: NOT_PERFORMED

[Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20260930-r06-combination-exact-count-0019.json)