The Mechanism · Gated-generation receipt

gen-define-permutations-and-combinations-and-give-one-short-work-fd99

stable pending witness
2026-05-29 02:48:41 UTC · anthropic/claude-sonnet-4-5

Prompt

Define permutations and combinations and give one short worked example.
71 chars

Generation

# Permutations and Combinations ## **Permutations** Arrangements where **order matters**. The number of ways to arrange *r* items from *n* items. **Formula:** P(n,r) = n!/(n-r)! ## **Combinations** Selections where **order doesn't matter**. The number of ways to choose *r* items from *n* items. **Formula:** C(n,r) = n!/[r!(n-r)!] --- ## **Worked Example** **Problem:** You have 5 books. How many ways can you: - a) Arrange 3 books on a shelf? - b) Choose 3 books to pack? **Solution:** **a) Permutation** (arrangement - order matters): - P(5,3) = 5!/(5-3)! = 5!/2! = (5×4×3×2×1)/(2×1) = **60 ways** **b) Combination** (selection - order doesn't matter): - C(5,3) = 5!/[3!(5-3)!] = 5!/(3!×2!) = (5×4×3!)/(3!×2×1) = **10 ways** --- **Key difference:** ABC, ACB, and BAC are different permutations but the same combination.
20 tokens in · 328 tokens out · 5563 ms · $0.005

The four gates

RED
pass
no disqualifying input patterns detected
FLOOR
pass
output above protective minimum; no mismatches
WAY
pass
no declared way_path (Way check NA); no coercion language
EXECUTION
deferred
no witnesses available (Witness Roll not yet constituted)

Verifier results

scripture_anchors NOT_APPLICABLE
0 Scripture citation(s) extracted
theology_doctrine NOT_APPLICABLE
no doctrine keyword patterns detected

Metrics

Total latency
5569.7 ms
Base LLM
5562.6 ms
Verifiers
0.2 ms
Gates
0.0 ms
Cost
$0.005

Trail

02:48:41 received prompt_chars=71 verifiers=['scripture_anchors', 'theology_doctrine']
02:48:41 red_gate decision=pass reason=no disqualifying input patterns detected
02:48:47 base_llm_call model=anthropic/claude-sonnet-4-5 tokens_in=20 tokens_out=328 latency_ms=5562.6 cost_usd=0.005
02:48:47 verifier:scripture_anchors verdict=NOT_APPLICABLE summary=0 Scripture citation(s) extracted
02:48:47 verifier:theology_doctrine verdict=NOT_APPLICABLE summary=no doctrine keyword patterns detected
02:48:47 floor_gate decision=pass reason=output above protective minimum; no mismatches
02:48:47 way_gate decision=pass reason=no declared way_path (Way check NA); no coercion language
02:48:47 execution_gate decision=deferred reason=no witnesses available (Witness Roll not yet constituted)
02:48:47 signed hash_algo=sha256 hash=sha256:82bdaac47ac548377ad490ea0603314a027c4a912d0ab818c579c969f01ad6e9

Content hash

sha256:82bdaac47ac548377ad490ea0603314a027c4a912d0ab818c579c969f01ad6e9

SHA256 over the canonical JSON (excluding this field). Tamper detection. Ed25519 signing planned for v2 once the operator’s signing key is provisioned on disk.

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