Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=1, k=1, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=0?).
G001 — Select distinct labeled objects without replacement. Ordered and unordered selections are different; the empty selection counts exactly once, with no circular or indistinguishable-object model. n=1, k=1, model=distinct_without_replacement. Give (P(n,k), C(n,k), k!, is k=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.
density14/15 use FULL-prefix underscores;density17 uses LOCAL hyphens.The coordinate and circle frames differ.Retain THIS family ordered answer roles/types and all original bilingual JSON/status.
Selection original full ordered ORIGINAL available-choice factor0 := mul(1, 1)
After
Selection original full ordered ORIGINAL available-choice factor0 = 1
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
source roles = [Selection original full ordered, 1, 1]
After
derived = [[1], 1]
Conditions: none
At position j there are n-j UNUSED distinct labeled objects.Multiply ALL k available-choice counts;at k0 there is exactly ONE empty ordered tuple.This is not replacement,circular or indistinguishable selection.
Selection original unordered ordered ORIGINAL available-choice factor0 := mul(1, 1)
After
Selection original unordered ordered ORIGINAL available-choice factor0 = 1
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
source roles = [Selection original unordered ordered, 1, 1]
After
derived = [[1], 1]
Conditions: none
At position j there are n-j UNUSED distinct labeled objects.Multiply ALL k available-choice counts;at k0 there is exactly ONE empty ordered tuple.This is not replacement,circular or indistinguishable selection.
Selection original unordered ORIGINAL exact ordered over k factorial := div(1, 1)
After
Selection original unordered ORIGINAL exact ordered over k factorial = 1
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
Selection original unordered reverse unordered times k factorial := mul(1, 1)
After
Selection original unordered reverse unordered times k factorial = 1
Conditions: none
Use ORIGINAL operands for exact integer/rational arithmetic.Only RAW inputs use SAFE or the explicit family bounds;derived values do not inherit those caps or use floating approximations.
source roles = [Selection original unordered, 1, 1, 1, 1]
After
derived = [1, 1]
Conditions: none
EACH k-subset has exactly k! orderings,so P/k! is the exact integer C.Multiply back to verify P.An empty subset and empty ordering each count once;no probability model is asserted.
S010 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S009 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S008 · premise_and_result_audit — Recompute Selection original unordered reverse unordered times k factorial with exact rational arithmetic, not rounded decimals.
S007 · premise_and_result_audit — Recompute Selection original unordered ORIGINAL exact ordered over k factorial with exact rational arithmetic, not rounded decimals.
S006 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S005 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S004 · premise_and_result_audit — Recompute Selection original unordered ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals.
S003 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
S002 · premise_and_result_audit — Recompute Selection original full ordered ORIGINAL available-choice factor0 with exact rational arithmetic, not rounded decimals.
S001 · premise_and_result_audit — Reverse-check using the ORIGINAL data,types,indices and premises.Consistency auditing does not claim the conclusion uniquely determines its inputs.Keep finite candidates,independent review,formal proof and publication separate.
Result
(1, 1, 1, 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.