ag-gipp/GoUldI

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data/202.json

Summary

Maintainability
Test Coverage
{
  "definitions" : {
    "k" : [ {
      "Q21199" : "natural number (numbers used for counting and ordering)"
    } ],
    "n" : [ {
      "Q21199" : "natural number (numbers used for counting and ordering)"
    } ]
  },
  "constraints" : [ ],
  "math_inputtex" : "|[k]^{n}(1\\text{-}1)|=\\binom{k}{n}n!=(k)_{n},",
  "math_inputtex_semantic" : "|[\\w{Q21199}{k}]^{\\w{Q21199}{n}}(1\\text{-}1)|=\\binom{k}{n}n!=(k)_{n}",
  "correct_tex" : "|[k]^{n}(1\\text{-}1)|=\\binom{k}{n}n!=(k)_{n}",
  "correct_mml" : "<math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"p1.1.m1.1\" class=\"ltx_Math\" alttext=\"|[k]^{n}(1\\text{-}1)|=\\binom{k}{n}n!=(k)_{n}\" display=\"inline\">\n  <semantics id=\"p1.1.m1.1a\">\n    <mrow id=\"p1.1.m1.1.21\" xref=\"p1.1.m1.1.21.cmml\">\n      <mrow id=\"p1.1.m1.1.21.2\" xref=\"p1.1.m1.1.21.2.1.cmml\">\n        <mo stretchy=\"false\" id=\"p1.1.m1.1.1\" xref=\"p1.1.m1.1.21.2.1.1.cmml\">|</mo>\n        <mrow id=\"p1.1.m1.1.21.2.2\" xref=\"p1.1.m1.1.21.2.2.cmml\">\n          <msup id=\"p1.1.m1.1.21.2.2.2\" xref=\"p1.1.m1.1.21.2.2.2.cmml\">\n            <mrow id=\"p1.1.m1.1.21.2.2.2.2\" xref=\"p1.1.m1.1.21.2.2.2.2.1.cmml\">\n              <mo stretchy=\"false\" id=\"p1.1.m1.1.2\" xref=\"p1.1.m1.1.21.2.2.2.2.1.1.cmml\">[</mo>\n              <mi id=\"p1.1.m1.1.3\" xref=\"p1.1.m1.1.3.cmml\">k</mi>\n              <mo stretchy=\"false\" id=\"p1.1.m1.1.4\" xref=\"p1.1.m1.1.21.2.2.2.2.1.1.cmml\">]</mo>\n            </mrow>\n            <mi id=\"p1.1.m1.1.5.1\" xref=\"p1.1.m1.1.5.1.cmml\">n</mi>\n          </msup>\n          <mo id=\"p1.1.m1.1.21.2.2.1\" xref=\"p1.1.m1.1.21.2.2.1.cmml\">⁢</mo>\n          <mrow id=\"p1.1.m1.1.21.2.2.3\" xref=\"p1.1.m1.1.21.2.2.3.2.cmml\">\n            <mo stretchy=\"false\" id=\"p1.1.m1.1.6\" xref=\"p1.1.m1.1.21.2.2.3.2.cmml\">(</mo>\n            <mrow id=\"p1.1.m1.1.21.2.2.3.2\" xref=\"p1.1.m1.1.21.2.2.3.2.cmml\">\n              <mn id=\"p1.1.m1.1.7\" xref=\"p1.1.m1.1.7.cmml\">1</mn>\n              <mo id=\"p1.1.m1.1.21.2.2.3.2.1\" xref=\"p1.1.m1.1.21.2.2.3.2.1.cmml\">⁢</mo>\n              <mtext id=\"p1.1.m1.1.8\" xref=\"p1.1.m1.1.8.cmml\">-</mtext>\n              <mo id=\"p1.1.m1.1.21.2.2.3.2.1a\" xref=\"p1.1.m1.1.21.2.2.3.2.1.cmml\">⁢</mo>\n              <mn id=\"p1.1.m1.1.9\" xref=\"p1.1.m1.1.9.cmml\">1</mn>\n            </mrow>\n            <mo stretchy=\"false\" id=\"p1.1.m1.1.10\" xref=\"p1.1.m1.1.21.2.2.3.2.cmml\">)</mo>\n          </mrow>\n        </mrow>\n        <mo stretchy=\"false\" id=\"p1.1.m1.1.11\" xref=\"p1.1.m1.1.21.2.1.1.cmml\">|</mo>\n      </mrow>\n      <mo id=\"p1.1.m1.1.12\" xref=\"p1.1.m1.1.12.cmml\">=</mo>\n      <mrow id=\"p1.1.m1.1.21.3\" xref=\"p1.1.m1.1.21.3.cmml\">\n        <mrow id=\"p1.1.m1.1.13.5\" xref=\"p1.1.m1.1.13.1.cmml\">\n          <mo id=\"p1.1.m1.1.13.2\" xref=\"p1.1.m1.1.13.1.1.cmml\">(</mo>\n          <mfrac linethickness=\"0pt\" id=\"p1.1.m1.1.13.3\" xref=\"p1.1.m1.1.13.1.cmml\">\n            <mi id=\"p1.1.m1.1.13.3.2\" xref=\"p1.1.m1.1.13.3.2.cmml\">k</mi>\n            <mi id=\"p1.1.m1.1.13.3.3\" xref=\"p1.1.m1.1.13.3.3.cmml\">n</mi>\n          </mfrac>\n          <mo id=\"p1.1.m1.1.13.4\" xref=\"p1.1.m1.1.13.1.1.cmml\">)</mo>\n        </mrow>\n        <mo id=\"p1.1.m1.1.21.3.1\" xref=\"p1.1.m1.1.21.3.1.cmml\">⁢</mo>\n        <mrow id=\"p1.1.m1.1.21.3.2\" xref=\"p1.1.m1.1.21.3.2.cmml\">\n          <mi id=\"p1.1.m1.1.14\" xref=\"p1.1.m1.1.14.cmml\">n</mi>\n          <mo lspace=\"0pt\" rspace=\"3.5pt\" id=\"p1.1.m1.1.15\" xref=\"p1.1.m1.1.15.cmml\">!</mo>\n        </mrow>\n      </mrow>\n      <mo id=\"p1.1.m1.1.16\" xref=\"p1.1.m1.1.16.cmml\">=</mo>\n      <msub id=\"p1.1.m1.1.21.4\" xref=\"p1.1.m1.1.21.4.cmml\">\n        <mrow id=\"p1.1.m1.1.21.4.2\" xref=\"p1.1.m1.1.21.4.cmml\">\n          <mo stretchy=\"false\" id=\"p1.1.m1.1.17\" xref=\"p1.1.m1.1.21.4.cmml\">(</mo>\n          <mi id=\"p1.1.m1.1.18\" xref=\"p1.1.m1.1.18.cmml\">k</mi>\n          <mo stretchy=\"false\" id=\"p1.1.m1.1.19\" xref=\"p1.1.m1.1.21.4.cmml\">)</mo>\n        </mrow>\n        <mi id=\"p1.1.m1.1.20.1\" xref=\"p1.1.m1.1.20.1.cmml\">n</mi>\n      </msub>\n    </mrow>\n    <annotation-xml encoding=\"MathML-Content\" id=\"p1.1.m1.1b\">\n      <apply id=\"p1.1.m1.1.21.cmml\" xref=\"p1.1.m1.1.21\">\n        <and id=\"p1.1.m1.1.21a.cmml\" xref=\"p1.1.m1.1.21\"/>\n        <apply id=\"p1.1.m1.1.21b.cmml\" xref=\"p1.1.m1.1.21\">\n          <eq id=\"p1.1.m1.1.12.cmml\" xref=\"p1.1.m1.1.12\"/>\n          <apply id=\"p1.1.m1.1.21.2.1.cmml\" xref=\"p1.1.m1.1.21.2\">\n            <abs id=\"p1.1.m1.1.21.2.1.1.cmml\" xref=\"p1.1.m1.1.1\"/>\n            <apply id=\"p1.1.m1.1.21.2.2.cmml\" xref=\"p1.1.m1.1.21.2.2\">\n              <times id=\"p1.1.m1.1.21.2.2.1.cmml\" xref=\"p1.1.m1.1.21.2.2.1\"/>\n              <apply id=\"p1.1.m1.1.21.2.2.2.cmml\" xref=\"p1.1.m1.1.21.2.2.2\">\n                <power  xref=\"p1.1.m1.1.21.2.2.2\"/>\n                <apply id=\"p1.1.m1.1.21.2.2.2.2.1.cmml\" xref=\"p1.1.m1.1.21.2.2.2.2\">\n                  <csymbol cd=\"latexml\" id=\"p1.1.m1.1.21.2.2.2.2.1.1.cmml\" xref=\"p1.1.m1.1.2\">delimited-[]</csymbol>\n                  <csymbol cd=\"latexml\" id=\"p1.1.m1.1.3.cmml\" xref=\"p1.1.m1.1.3\">Q21199</csymbol>\n                </apply>\n                <csymbol cd=\"latexml\" id=\"p1.1.m1.1.5.1.cmml\" xref=\"p1.1.m1.1.5.1\">Q21199</csymbol>\n              </apply>\n              <apply id=\"p1.1.m1.1.21.2.2.3.2.cmml\" xref=\"p1.1.m1.1.21.2.2.3\">\n                <times id=\"p1.1.m1.1.21.2.2.3.2.1.cmml\" xref=\"p1.1.m1.1.21.2.2.3.2.1\"/>\n                <cn type=\"integer\" id=\"p1.1.m1.1.7.cmml\" xref=\"p1.1.m1.1.7\">1</cn>\n                <mtext id=\"p1.1.m1.1.8.cmml\" xref=\"p1.1.m1.1.8\">-</mtext>\n                <cn type=\"integer\" id=\"p1.1.m1.1.9.cmml\" xref=\"p1.1.m1.1.9\">1</cn>\n              </apply>\n            </apply>\n          </apply>\n          <apply id=\"p1.1.m1.1.21.3.cmml\" xref=\"p1.1.m1.1.21.3\">\n            <times id=\"p1.1.m1.1.21.3.1.cmml\" xref=\"p1.1.m1.1.21.3.1\"/>\n            <apply id=\"p1.1.m1.1.13.1.cmml\" xref=\"p1.1.m1.1.13.5\">\n              <csymbol cd=\"latexml\" id=\"p1.1.m1.1.13.1.1.cmml\" xref=\"p1.1.m1.1.13.2\">binomial</csymbol>\n              <csymbol cd=\"latexml\" id=\"p1.1.m1.1.13.3.2.cmml\" xref=\"p1.1.m1.1.13.3.2\">Q21199</csymbol>\n              <csymbol cd=\"latexml\" id=\"p1.1.m1.1.13.3.3.cmml\" xref=\"p1.1.m1.1.13.3.3\">Q21199</csymbol>\n            </apply>\n            <apply id=\"p1.1.m1.1.21.3.2.cmml\" xref=\"p1.1.m1.1.21.3.2\">\n              <factorial id=\"p1.1.m1.1.15.cmml\" xref=\"p1.1.m1.1.15\"/>\n              <csymbol cd=\"latexml\" id=\"p1.1.m1.1.14.cmml\" xref=\"p1.1.m1.1.14\">Q21199</csymbol>\n            </apply>\n          </apply>\n        </apply>\n        <apply id=\"p1.1.m1.1.21c.cmml\" xref=\"p1.1.m1.1.21\">\n          <eq id=\"p1.1.m1.1.16.cmml\" xref=\"p1.1.m1.1.16\"/>\n          <share href=\"#p1.1.m1.1.21.3.cmml\" id=\"p1.1.m1.1.21d.cmml\" xref=\"p1.1.m1.1.21\"/>\n          <apply id=\"p1.1.m1.1.21.4.cmml\" xref=\"p1.1.m1.1.21.4\">\n            <csymbol cd=\"ambiguous\" id=\"p1.1.m1.1.21.4.1.cmml\" xref=\"p1.1.m1.1.21.4\">subscript</csymbol>\n            <csymbol cd=\"latexml\" id=\"p1.1.m1.1.18.cmml\" xref=\"p1.1.m1.1.18\">Q21199</csymbol>\n            <csymbol cd=\"latexml\" id=\"p1.1.m1.1.20.1.cmml\" xref=\"p1.1.m1.1.20.1\">Q21199</csymbol>\n          </apply>\n        </apply>\n      </apply>\n    </annotation-xml>\n    <annotation encoding=\"application/x-tex\" id=\"p1.1.m1.1c\">|[k]^{n}(1\\text{-}1)|=\\binom{k}{n}n!=(k)_{n}</annotation>\n  </semantics>\n</math>",
  "uri" : "https://arxiv.org/abs/math/0112281",
  "title" : "Factorial Product",
  "comment" : "",
  "type" : "definition",
  "ntcir12-type" : "2",
  "formula" : "1_8.4",
  "page" : "2",
  "ntcir12-relevance" : "3"
}