{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Proof\n\n1. The integers $\\Z$ form a commutative ring. Apply the [[binomial theorem|0EHWPWGHB2]] in $\\Z$ with $x = y = 1$.\n2. It gives $(1 + 1)^n = \\sum_{k=0}^{n} \\binom{n}{k} 1^k 1^{n-k}$.\n3. Every power of $1$ equals $1$, so the right side is $\\sum_{k=0}^{n} \\binom{n}{k}$.\n4. The left side is $2^n$. Hence $\\sum_{k=0}^{n} \\binom{n}{k} = 2^n$.\n\n## Idea\n\nThe binomial theorem with both variables equal to $1$ removes the monomials and leaves only the coefficients.\nTheir sum is then $(1 + 1)^n = 2^n$.\n",
  "commit": "4a3290d299c8b2cd6493fce2296920ba7f82d942",
  "front": {
    "kind": "proof",
    "msc": [
      "05A10"
    ],
    "name": "Proof of the sum of binomial coefficients from the binomial theorem",
    "proves": "0KAMF19NV8",
    "sources": [
      {
        "url": "https://proofwiki.org/wiki/Sum_of_Binomial_Coefficients_over_Lower_Index"
      }
    ],
    "tag": "GS2SF16HQY",
    "uses": [
      "0EHWPWGHB2",
      "092GEF38YD"
    ]
  },
  "kind": "proof",
  "link": "https://math2.tamnd.com/node/GS2SF16HQY/",
  "links": [
    {
      "field": "proves",
      "target": "0KAMF19NV8"
    },
    {
      "field": "uses",
      "target": "0EHWPWGHB2"
    },
    {
      "field": "uses",
      "target": "092GEF38YD"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/GS2SF16HQY.md",
  "msc": [
    "05A10"
  ],
  "name": "Proof of the sum of binomial coefficients from the binomial theorem",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GS2/GS2SF16HQY.md",
  "status": null,
  "tag": "GS2SF16HQY",
  "uri": "m2://GS2SF16HQY",
  "url": "https://math2.tamnd.com/wiki/proof-of-the-sum-of-binomial-coefficients-from-the-binomial-theorem/",
  "writing": {
    "tag": "GS2SF16HQY",
    "kind": "proof",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}