{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. Let $n$ and $k$ be natural numbers.\n2. The [[binomial coefficient|092GEF38YD]] is defined by the recursion $\\binom{n}{0} = 1$, $\\binom{0}{k+1} = 0$ and $\\binom{n+1}{k+1} = \\binom{n}{k} + \\binom{n}{k+1}$.\n3. The third clause of this definition, applied to $n$ and $k$, is the equation $\\binom{n+1}{k+1} = \\binom{n}{k} + \\binom{n}{k+1}$.\n\n## Idea\n\nIn this wiki the binomial coefficients are defined by Pascal's rule itself, so the rule holds by definition.\nWith other definitions, such as the factorial formula or the count of subsets, it needs a short computation or a counting argument.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "05A10"
    ],
    "name": "Proof of Pascal's rule from the recursive definition",
    "proves": "0YRG241HQQ",
    "sources": [
      {
        "locator": "lead section",
        "url": "https://en.wikipedia.org/wiki/Pascal's_rule"
      }
    ],
    "tag": "EDF0ZJF141",
    "uses": [
      "092GEF38YD"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/EDF0ZJF141/",
  "links": [
    {
      "field": "proves",
      "target": "0YRG241HQQ"
    },
    {
      "field": "uses",
      "target": "092GEF38YD"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/EDF0ZJF141.md",
  "msc": [
    "05A10"
  ],
  "name": "Proof of Pascal's rule from the recursive definition",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/EDF/EDF0ZJF141.md",
  "status": null,
  "tag": "EDF0ZJF141",
  "uri": "m2://EDF0ZJF141",
  "url": "https://math2.tamnd.com/wiki/proof-of-pascals-rule-from-the-recursive-definition/",
  "writing": {
    "tag": "EDF0ZJF141",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}