Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Multinomial




Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Multinomial[n1,n2,...,nm] > Summation > Finite summation





http://functions.wolfram.com/06.04.23.0001.01









  


  








Input Form





BoxData[\(\(\(Sum[\(\(Sum[\(\(Product[\(\(Multinomial[\(\(Subscript[\(k, i, 1\)]\), …, \(Subscript[\(k, i, n\)]\)\)]\), \({i, 1, m}\)\)]\), \({\(Subscript[\(k, m, n\)]\), 0, o}\)\)]\), …, \({\(Subscript[\(k, 1, 1\)]\), 0, o}\)\)]\)  \(Multinomial[\(\(Subscript[\(b, 1\)]\), …, \(Subscript[\(b, n\)]\)\)]\)\)/;\((\(\(\(Sum[\(\(Subscript[\(k, i, j\)]\), \({j, 1, n}\)\)]\)  \(Subscript[\(a, i\)]\)\) && \(\(Sum[\(\(Subscript[\(k, i, j\)]\), \({i, 1, m}\)\)]\)  \(Subscript[\(b, i\)]\)\) && \(\(Sum[\(\(Subscript[\(a, i\)]\), \({i, 1, m}\)\)]\)  \(Sum[\(\(Subscript[\(b, j\)]\), \({j, 1, n}\)\)]\)\) && o\) = \(Max[\(\(Subscript[\(k, 1, 1\)]\), …, \(Subscript[\(k, m, n\)]\)\)]\))\)\)]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", RowBox[List["1", ",", "1"]]], "=", "0"]], "o"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", RowBox[List["m", ",", "n"]]], "=", "0"]], "o"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "1"]], "m"], RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["k", RowBox[List["i", ",", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["k", RowBox[List["i", ",", "n"]]]]], "]"]]]]]]]]]], "\[Equal]", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "n"]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], SubscriptBox["k", RowBox[List["i", ",", "j"]]]]], "\[Equal]", SubscriptBox["a", "i"]]], "\[And]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "m"], SubscriptBox["k", RowBox[List["i", ",", "j"]]]]], "\[Equal]", SubscriptBox["b", "i"]]], "\[And]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "m"], SubscriptBox["a", "i"]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], SubscriptBox["b", "j"]]]]], "\[And]", "o"]]]], "=", RowBox[List["Max", "[", RowBox[List[SubscriptBox["k", RowBox[List["1", ",", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["k", RowBox[List["m", ",", "n"]]]]], "]"]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> o </mi> </munderover> <mrow> <mo> &#8230; </mo> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <msub> <mi> k </mi> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> o </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> k </mi> <mrow> <mi> i </mi> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> <mo> + </mo> <mo> &#8230; </mo> <mo> + </mo> <msub> <mi> k </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> ; </mo> <msub> <mi> k </mi> <mrow> <mi> i </mi> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> k </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[RowBox[List[SubscriptBox[&quot;k&quot;, RowBox[List[&quot;i&quot;, &quot;,&quot;, &quot;1&quot;]]], &quot;+&quot;, &quot;\[Ellipsis]&quot;, &quot;+&quot;, SubscriptBox[&quot;k&quot;, RowBox[List[&quot;i&quot;, &quot;,&quot;, &quot;n&quot;]]]]], &quot;;&quot;, SubscriptBox[&quot;k&quot;, RowBox[List[&quot;i&quot;, &quot;,&quot;, &quot;1&quot;]]]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, SubscriptBox[&quot;k&quot;, RowBox[List[&quot;i&quot;, &quot;,&quot;, &quot;n&quot;]]]]], &quot;)&quot;]], Multinomial] </annotation> </semantics> </mrow> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <semantics> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <msub> <mi> b </mi> <mn> 1 </mn> </msub> <mo> + </mo> <mo> &#8230; </mo> <mo> + </mo> <msub> <mi> b </mi> <mi> n </mi> </msub> </mrow> <mo> ; </mo> <msub> <mi> b </mi> <mn> 1 </mn> </msub> </mrow> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> b </mi> <mi> n </mi> </msub> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, RowBox[List[RowBox[List[RowBox[List[SubscriptBox[&quot;b&quot;, &quot;1&quot;], &quot;+&quot;, &quot;\[Ellipsis]&quot;, &quot;+&quot;, SubscriptBox[&quot;b&quot;, &quot;n&quot;]]], &quot;;&quot;, SubscriptBox[&quot;b&quot;, &quot;1&quot;]]], &quot;,&quot;, &quot;\[Ellipsis]&quot;, &quot;,&quot;, SubscriptBox[&quot;b&quot;, &quot;n&quot;]]], &quot;)&quot;]], Multinomial] </annotation> </semantics> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> k </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> </mrow> </msub> </mrow> <mo> &#10869; </mo> <msub> <mi> a </mi> <mi> i </mi> </msub> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> k </mi> <mrow> <mi> i </mi> <mo> , </mo> <mi> j </mi> </mrow> </msub> </mrow> <mo> &#10869; </mo> <msub> <mi> b </mi> <mi> i </mi> </msub> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <msub> <mi> a </mi> <mi> i </mi> </msub> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msub> <mi> b </mi> <mi> j </mi> </msub> </mrow> </mrow> <mo> &#8743; </mo> <mi> o </mi> </mrow> </mrow> <mo> = </mo> <mrow> <mi> Max </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> k </mi> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <msub> <mi> k </mi> <mrow> <mi> m </mi> <mo> , </mo> <mi> n </mi> </mrow> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Set </ci> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> <cn type='integer'> 1 </cn> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> o </ci> </uplimit> <apply> <times /> <ci> &#8230; </ci> <apply> <sum /> <bvar> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> m </ci> <ci> n </ci> </apply> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> o </ci> </uplimit> <apply> <product /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <ci> Multinomial </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> i </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Multinomial </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <cn type='integer'> 1 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> i </ci> <ci> j </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> i </ci> </apply> </apply> <apply> <eq /> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> i </ci> <ci> j </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> i </ci> </apply> </apply> <apply> <eq /> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> m </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> i </ci> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> j </ci> </apply> </apply> </apply> <ci> o </ci> </apply> </apply> <apply> <max /> <apply> <ci> Subscript </ci> <ci> k </ci> <cn type='integer'> 1 </cn> <cn type='integer'> 1 </cn> </apply> <ci> &#8230; </ci> <apply> <ci> Subscript </ci> <ci> k </ci> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", RowBox[List["1", ",", "1"]]], "=", "0"]], "o"], RowBox[List["\[Ellipsis]", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List[SubscriptBox["k", RowBox[List["m", ",", "n"]]], "=", "0"]], "o"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "1"]], "m"], RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["k", RowBox[List["i", ",", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["k", RowBox[List["i", ",", "n"]]]]], "]"]]]]]]]]]], "\[Equal]", RowBox[List["Multinomial", "[", RowBox[List[SubscriptBox["b", "1"], ",", "\[Ellipsis]", ",", SubscriptBox["b", "n"]]], "]"]]]], "/;", RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], SubscriptBox["k", RowBox[List["i", ",", "j"]]]]], "\[Equal]", SubscriptBox["a", "i"]]], "&&", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "m"], SubscriptBox["k", RowBox[List["i", ",", "j"]]]]], "\[Equal]", SubscriptBox["b", "i"]]], "&&", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "1"]], "m"], SubscriptBox["a", "i"]]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], SubscriptBox["b", "j"]]]]], "&&", "o"]]]], "=", RowBox[List["Max", "[", RowBox[List[SubscriptBox["k", RowBox[List["1", ",", "1"]]], ",", "\[Ellipsis]", ",", SubscriptBox["k", RowBox[List["m", ",", "n"]]]]], "]"]]]]]]


Date Added to functions.wolfram.com (modification date)





2002-12-18