{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nOne can generalize Vandermonde's identity as follows:\n\n$$\\sum_{k_1+\\cdots +k_p = m} {n_1\\choose k_1} {n_2\\choose k_2} \\cdots {n_p\\choose k_p} = { n_1+\\dots +n_p \\choose m }.$$\n\nThis identity can be obtained through the algebraic derivation above when more than two polynomials are used, or through a simple double counting argument.\n\nOn the one hand, one chooses $\\textstyle k_1$ elements out of a first set of $\\textstyle n_1$ elements; then $\\textstyle k_2$ out of another set, and so on, through $\\textstyle p$ such sets, until a total of $\\textstyle m$ elements have been chosen from the $\\textstyle p$ sets. One therefore chooses $\\textstyle m$ elements out of $\\textstyle n_1+\\dots +n_p$ in the left-hand side, which is also exactly what is done in the right-hand side.\n",
  "commit": "68a8c3dc73fd55004b2796431c607c30960d1a85",
  "front": {
    "inventory": [
      "ZXMN45KDPR"
    ],
    "kind": "statement",
    "msc": [
      "05A19"
    ],
    "name": "Generalized Vandermonde's identity",
    "role": "proposition",
    "sources": [
      {
        "locator": "Vandermonde's identity, section Generalizations \u003e Generalized Vandermonde's identity",
        "mention": "wikipedia://en/916157#MBB0",
        "note": "adopted",
        "sections": [
          "Statement"
        ],
        "wikipedia": "en:916157@r1370490483"
      }
    ],
    "tag": "6RRNPDPW91"
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/6RRNPDPW91/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/6RRNPDPW91.md",
  "msc": [
    "05A19"
  ],
  "name": "Generalized Vandermonde's identity",
  "prov": null,
  "role": "proposition",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6RR/6RRNPDPW91.md",
  "status": null,
  "tag": "6RRNPDPW91",
  "uri": "m2://6RRNPDPW91",
  "url": "https://math2.tamnd.com/wiki/generalized-vandermondes-identity/",
  "writing": {
    "tag": "6RRNPDPW91",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "edge uses out",
        "note": "each definition that the statement needs",
        "have": 0,
        "want": 1
      }
    ]
  }
}