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Question 34196: Exact Ordered Selection Count

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

R0067-34196 · q-20260930-r06-permutation-exact-count-0021

Original question and givens

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

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

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.

Step-by-step derivation

S001 · implies

Before

source roles = [qf-permutation-exact-count, [k, n, operation]]

After

derived = [permutation, no-invented-domain]

Conditions: none

Check THIS bank’s keys/operation/domain:legacy table underscores,MOD50 local hyphens;permissive BASE data acquires no unknown-field restriction.Keep Number.isInteger and SAFE source guards distinct,while preserving ORIGINAL JSON number representations.

Step identifiers and dependencies

Rule: R67_SOURCE_specific_keys_operation_domain_and_numeric_bank · Uses: G001

S002 · implies

Before

source roles = [8, 8]

After

derived = [[8, 7, 6, 5, 4, 3, 2, 1], 1]

Conditions: none

Choose ordered positions from n DISTINCT objects WITHOUT replacement.Position i has n−i+1 choices.For k=0,the empty selection counts ONCE and the product starts at1.THIS bank permits n≤200,not another bank’s32/64 cap.

Step identifiers and dependencies

Rule: Selection_DISTINCT_no_replacement_order_and_EMPTY_convention · Uses: G001, S001

S003 · calculation

Before

Permutation ORIGINAL available choices position1 := mul(1, 8)

After

Permutation ORIGINAL available choices position1 = 8

Conditions: none

Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

Permutation ORIGINAL available choices position2 := mul(8, 7)

After

Permutation ORIGINAL available choices position2 = 56

Conditions: none

Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

Permutation ORIGINAL available choices position3 := mul(56, 6)

After

Permutation ORIGINAL available choices position3 = 336

Conditions: none

Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

Permutation ORIGINAL available choices position4 := mul(336, 5)

After

Permutation ORIGINAL available choices position4 = 1680

Conditions: none

Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

Permutation ORIGINAL available choices position5 := mul(1680, 4)

After

Permutation ORIGINAL available choices position5 = 6720

Conditions: none

Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

Permutation ORIGINAL available choices position6 := mul(6720, 3)

After

Permutation ORIGINAL available choices position6 = 20160

Conditions: none

Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

Permutation ORIGINAL available choices position7 := mul(20160, 2)

After

Permutation ORIGINAL available choices position7 = 40320

Conditions: none

Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · calculation

Before

Permutation ORIGINAL available choices position8 := mul(40320, 1)

After

Permutation ORIGINAL available choices position8 = 40320

Conditions: none

Calculate exactly from ORIGINAL ordered operands.Legacy derived integers are not recapped at the RAW SAFE bound;MOD50 rationals separately retain the NAIVE pre-reduction source envelope.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S009

S011 · implies

Before

source roles = [8, 8]

After

derived = [40320]

Conditions: none

The multiplication principle combines ALL k position counts,giving P(n,k)=n!/(n−k)!.This is a count,not probability.The empty product is1 and derived BigInts are not truncated to SAFE.

Step identifiers and dependencies

Rule: Selection_ALL_ordered_choices_falling_product · Uses: G001, S010

Reverse checks

  • S011 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
  • S010 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position8 with exact rational arithmetic, not rounded decimals.
  • S009 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position7 with exact rational arithmetic, not rounded decimals.
  • S008 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position6 with exact rational arithmetic, not rounded decimals.
  • S007 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position5 with exact rational arithmetic, not rounded decimals.
  • S006 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position4 with exact rational arithmetic, not rounded decimals.
  • S005 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position3 with exact rational arithmetic, not rounded decimals.
  • S004 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position2 with exact rational arithmetic, not rounded decimals.
  • S003 · premise_and_result_audit — Recompute Permutation ORIGINAL available choices position1 with exact rational arithmetic, not rounded decimals.
  • S002 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.
  • S001 · premise_and_result_audit — Reverse-audit THIS step against the ORIGINAL roles,complete premises and field order.Distinguish implication from equivalence;preserve originals,English,MKOs,ASTs and review states.Finite checks are not independent review or general formal proof.

Result

40320

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: 93ba60ceba7f65b8d5ab485b7417fdecbe986bd313a46099c80a1e82bdbe4e40
  • candidate record SHA-256: ea20f3bf7ee64bfcd58b7b6051e36bfa00310eb548a92f2f32e8e0c630d54acc
  • semantic review receipt SHA-256: c89efa383b67bb2cf25bcf16160b9f5f614bbe5e5e791f4d7f58234d17a3371c
  • question-first witness SHA-256: 12766d451c5375842b4fc2843ae94807ec55f15763a916558026912a13873d57
  • Lean: NOT_RUN · human review: NOT_PERFORMED
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