{"id": "card_works_counting", "kind": "verified", "title": "Counting: how many ways to choose and to arrange", "body": "From five things, there are 20 ordered pairs but only 10 unordered pairs — order is the whole difference between a permutation and a combination.  Worked & sealed by the engine — P(5,2) = 5!/3! = 20 · C(5,2) = 10.  [HOLDS; open the seal to re-check.]", "source": {"label": "The Works — worked & sealed", "url": "https://narrowhighway.com/s/b7807a96bdad28ab5ec46b81440bc5ce34b7f534ac40a1b639088ba64de05751", "domain": "combinatorics", "authority_tier": "verified"}, "shelf": "the-works", "box": "mathematics", "bands": ["the-works", "mathematics", "combinatorics", "combinatorics"], "connections": [{"to_card_id": "card_spine_the-works", "relationship": "member_of", "evidence": "a member of the the-works shelf in the keeping"}, {"to_card_id": "card_k_floor_of_discovery", "relationship": "paves", "evidence": "A worked, engine-sealed demonstration — a paving-stone of the floor of reality."}, {"to_card_id": "card_works_number_theory", "relationship": "adjacent", "evidence": "Neighbouring worked demonstrations in the same part of the volume."}, {"to_card_id": "card_works_euler_identity", "relationship": "adjacent", "evidence": "Neighbouring worked demonstrations in the same part of the volume."}], "author": "engine", "created_at": 0.0, "updated_at": 0.0, "visibility": "public", "lifecycle_stage": "public", "volatility": "permanent", "surface": "secular", "generated": false, "subject": "Combinatorics", "extra": {"seal_hash": "b7807a96bdad28ab5ec46b81440bc5ce34b7f534ac40a1b639088ba64de05751", "cite_url": "https://narrowhighway.com/s/b7807a96bdad28ab5ec46b81440bc5ce34b7f534ac40a1b639088ba64de05751", "demonstration_id": "works_counting", "verdict": "HOLDS"}, "presentation": {"glyph": "•", "kind_label": "mathematics", "by": "The Works — worked & sealed", "authority": "verified", "posted": "", "standing": "", "link": {"url": "https://narrowhighway.com/s/b7807a96bdad28ab5ec46b81440bc5ce34b7f534ac40a1b639088ba64de05751", "host": "narrowhighway.com", "provider": "", "titled": "Counting: how many ways to choose and to arrange", "waybill_line": "", "reach": "", "embed": ""}}, "neighbors": [{"id": "card_spine_the-works", "title": "The Works — sealed, worked demonstrations", "relationship": "on the shelf of", "why": "a member of the the-works shelf in the keeping", "href": "/card/card_spine_the-works", "resolved": true}, {"id": "card_k_floor_of_discovery", "title": "The Floor of Discovery — one floor, and by its design the fear of God", "relationship": "paves", "why": "A worked, engine-sealed demonstration — a paving-stone of the floor of reality.", "href": "/card/card_k_floor_of_discovery", "resolved": true}, {"id": "card_works_number_theory", "title": "The integers, examined: a prime, a divisor, a factorial", "relationship": "adjacent", "why": "Neighbouring worked demonstrations in the same part of the volume.", "href": "/card/card_works_number_theory", "resolved": true}, {"id": "card_works_euler_identity", "title": "Euler's identity, assembled from its parts", "relationship": "adjacent", "why": "Neighbouring worked demonstrations in the same part of the volume.", "href": "/card/card_works_euler_identity", "resolved": true}]}