@prefix psr: <http://data.loterre.fr/ark:/67375/PSR> .
@prefix skos: <http://www.w3.org/2004/02/skos/core#> .
@prefix dc: <http://purl.org/dc/terms/> .
@prefix xsd: <http://www.w3.org/2001/XMLSchema#> .

psr:-W96QGKZX-0
  skos:prefLabel "elementary algebra"@en, "algèbre élémentaire"@fr ;
  a skos:Concept ;
  skos:narrower psr:-G8L5RZH2-4 .

psr:-F7H3K8H1-0
  skos:prefLabel "coefficient binomial"@fr, "binomial coefficient"@en ;
  a skos:Concept ;
  skos:related psr:-G8L5RZH2-4 .

psr: a skos:ConceptScheme .
psr:-G8L5RZH2-4
  skos:definition """La formule du binôme de Newton est une formule mathématique donnée par Isaac Newton pour trouver le développement d'une puissance entière quelconque d'un binôme. Elle est aussi appelée formule du binôme ou formule de Newton. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Formule_du_bin%C3%B4me_de_Newton">https://fr.wikipedia.org/wiki/Formule_du_bin%C3%B4me_de_Newton</a>)"""@fr, """In elementary algebra, the <b>binomial theorem</b> (or <b>binomial expansion</b>) describes the algebraic expansion of powers of a binomial. According to the theorem, it is possible to expand the polynomial <span class="texhtml">(<i>x</i> + <i>y</i>)<sup><i>n</i></sup></span> into a sum involving terms of the form <span class="texhtml"><i>ax</i><sup><i>b</i></sup><i>y</i><sup><i>c</i></sup></span>, where the exponents <span class="texhtml mvar" style="font-style:italic;">b</span> and <span class="texhtml mvar" style="font-style:italic;">c</span> are nonnegative integers with <span class="texhtml"><i>b</i> + <i>c</i> = <i>n</i></span>, and the coefficient <span class="texhtml mvar" style="font-style:italic;">a</span> of each term is a specific positive integer depending on <span class="texhtml mvar" style="font-style:italic;">n</span> and <span class="texhtml mvar" style="font-style:italic;">b</span>.  For example, for <span class="texhtml"><i>n</i> = 4</span>, <div class="mwe-math-element"><div class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mo stretchy="false">(</mo>         <mi>x</mi>         <mo>+</mo>         <mi>y</mi>         <msup>           <mo stretchy="false">)</mo>           <mrow class="MJX-TeXAtom-ORD">             <mn>4</mn>           </mrow>         </msup>         <mo>=</mo>         <msup>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>4</mn>           </mrow>         </msup>         <mo>+</mo>         <mn>4</mn>         <msup>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>3</mn>           </mrow>         </msup>         <mi>y</mi>         <mo>+</mo>         <mn>6</mn>         <msup>           <mi>x</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <msup>           <mi>y</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>2</mn>           </mrow>         </msup>         <mo>+</mo>         <mn>4</mn>         <mi>x</mi>         <msup>           <mi>y</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>3</mn>           </mrow>         </msup>         <mo>+</mo>         <msup>           <mi>y</mi>           <mrow class="MJX-TeXAtom-ORD">             <mn>4</mn>           </mrow>         </msup>         <mo>.</mo>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}</annotation>   </semantics> </math></div><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd0ade2ebe2cf6c1b2a7084b3b5cbf0b8e3a77b" class="mwe-math-fallback-image-display mw-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.064ex; height:3.176ex;" alt="{\\\\displaystyle (x+y)^{4}=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4}.}"></div> The coefficient <span class="texhtml mvar" style="font-style:italic;">a</span> in the term of <span class="texhtml"><i>ax</i><sup><i>b</i></sup><i>y</i><sup><i>c</i></sup></span> is known as the binomial coefficient <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\	binom {n}{b}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mstyle displaystyle="false" scriptlevel="0">             <mrow>               <mrow class="MJX-TeXAtom-OPEN">                 <mo maxsize="1.2em" minsize="1.2em">(</mo>               </mrow>               <mfrac linethickness="0">                 <mi>n</mi>                 <mi>b</mi>               </mfrac>               <mrow class="MJX-TeXAtom-CLOSE">                 <mo maxsize="1.2em" minsize="1.2em">)</mo>               </mrow>             </mrow>           </mstyle>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\	binom {n}{b}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39fb0660ed38bb67c05b149fc8cfa90b3a48fc20" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\\\\displaystyle {\\	binom {n}{b}}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\	binom {n}{c}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mstyle displaystyle="false" scriptlevel="0">             <mrow>               <mrow class="MJX-TeXAtom-OPEN">                 <mo maxsize="1.2em" minsize="1.2em">(</mo>               </mrow>               <mfrac linethickness="0">                 <mi>n</mi>                 <mi>c</mi>               </mfrac>               <mrow class="MJX-TeXAtom-CLOSE">                 <mo maxsize="1.2em" minsize="1.2em">)</mo>               </mrow>             </mrow>           </mstyle>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\	binom {n}{c}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0325ad33b80d9aee734b6a82defe6e5971ec5763" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\\\\displaystyle {\\	binom {n}{c}}}"></span> (the two have the same value). These coefficients for varying <span class="texhtml mvar" style="font-style:italic;">n</span> and <span class="texhtml mvar" style="font-style:italic;">b</span> can be arranged to form Pascal's triangle. These numbers also occur in combinatorics, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\	binom {n}{b}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mstyle displaystyle="false" scriptlevel="0">             <mrow>               <mrow class="MJX-TeXAtom-OPEN">                 <mo maxsize="1.2em" minsize="1.2em">(</mo>               </mrow>               <mfrac linethickness="0">                 <mi>n</mi>                 <mi>b</mi>               </mfrac>               <mrow class="MJX-TeXAtom-CLOSE">                 <mo maxsize="1.2em" minsize="1.2em">)</mo>               </mrow>             </mrow>           </mstyle>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\	binom {n}{b}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39fb0660ed38bb67c05b149fc8cfa90b3a48fc20" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\\\\displaystyle {\\	binom {n}{b}}}"></span> gives the number of different combinations of <span class="texhtml mvar" style="font-style:italic;">b</span> elements that can be chosen from an <span class="texhtml mvar" style="font-style:italic;">n</span>-element set.  Therefore <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\\\\displaystyle {\\	binom {n}{b}}}">   <semantics>     <mrow class="MJX-TeXAtom-ORD">       <mstyle displaystyle="true" scriptlevel="0">         <mrow class="MJX-TeXAtom-ORD">           <mstyle displaystyle="false" scriptlevel="0">             <mrow>               <mrow class="MJX-TeXAtom-OPEN">                 <mo maxsize="1.2em" minsize="1.2em">(</mo>               </mrow>               <mfrac linethickness="0">                 <mi>n</mi>                 <mi>b</mi>               </mfrac>               <mrow class="MJX-TeXAtom-CLOSE">                 <mo maxsize="1.2em" minsize="1.2em">)</mo>               </mrow>             </mrow>           </mstyle>         </mrow>       </mstyle>     </mrow>     <annotation encoding="application/x-tex">{\\\\displaystyle {\\	binom {n}{b}}}</annotation>   </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39fb0660ed38bb67c05b149fc8cfa90b3a48fc20" class="mwe-math-fallback-image-inline mw-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.116ex; height:3.176ex;" alt="{\\\\displaystyle {\\	binom {n}{b}}}"></span> is often pronounced as "<span class="texhtml mvar" style="font-style:italic;">n</span> choose <span class="texhtml mvar" style="font-style:italic;">b</span>". 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Binomial_theorem">https://en.wikipedia.org/wiki/Binomial_theorem</a>)"""@en ;
  skos:altLabel "binomial expansion"@en ;
  skos:exactMatch <https://en.wikipedia.org/wiki/Binomial_theorem>, <https://fr.wikipedia.org/wiki/Formule_du_bin%C3%B4me_de_Newton> ;
  skos:related psr:-F7H3K8H1-0 ;
  skos:inScheme psr: ;
  dc:modified "2024-10-18"^^xsd:date ;
  skos:broader psr:-W96QGKZX-0 ;
  a skos:Concept ;
  dc:created "2023-07-13"^^xsd:date ;
  skos:prefLabel "formule du binôme de Newton"@fr, "binomial theorem"@en .