@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:-B373Q2P1-V
  skos:prefLabel "combinatorics"@en, "combinatoire"@fr ;
  a skos:Concept ;
  skos:narrower psr:-PJ28VRBP-W .

psr: a skos:ConceptScheme .
psr:-VTR5XXB2-M
  skos:prefLabel "identité"@fr, "identity"@en ;
  a skos:Concept ;
  skos:narrower psr:-PJ28VRBP-W .

psr:-PJ28VRBP-W
  skos:altLabel "Vandermonde's convolution"@en, "formule de convolution"@fr ;
  skos:prefLabel "identité de Vandermonde"@fr, "Vandermonde's identity"@en ;
  skos:broader psr:-B373Q2P1-V, psr:-VTR5XXB2-M ;
  skos:inScheme psr: ;
  a skos:Concept ;
  skos:exactMatch <https://fr.wikipedia.org/wiki/Identit%C3%A9_de_Vandermonde>, <https://en.wikipedia.org/wiki/Vandermonde%27s_identity> ;
  skos:definition """In combinatorics, <b>Vandermonde's identity</b> (or <b>Vandermonde's convolution</b>) is the following identity for binomial coefficients:
<br/>
<br/><dl><dd><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 {m+n \\\\choose r}=\\\\sum _{k=0}^{r}{m \\\\choose k}{n \\\\choose r-k}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mrow>
<br/>                <mi>m</mi>
<br/>                <mo>+</mo>
<br/>                <mi>n</mi>
<br/>              </mrow>
<br/>              <mi>r</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>r</mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mi>m</mi>
<br/>              <mi>k</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mi>n</mi>
<br/>              <mrow>
<br/>                <mi>r</mi>
<br/>                <mo>−<!-- − --></mo>
<br/>                <mi>k</mi>
<br/>              </mrow>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {m+n \\\\choose r}=\\\\sum _{k=0}^{r}{m \\\\choose k}{n \\\\choose r-k}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdad4f9c347f675b3d24ba5cd33a46f250a17f8f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.171ex; width:30.52ex; height:7.009ex;" alt="{\\\\displaystyle {m+n \\\\choose r}=\\\\sum _{k=0}^{r}{m \\\\choose k}{n \\\\choose r-k}}"></span></dd></dl>
<br/>for any nonnegative integers <i>r</i>, <i>m</i>, <i>n</i>.  The identity is named after Alexandre-Théophile Vandermonde (1772), although it was already known in 1303 by the Chinese mathematician Zhu Shijie.
<br/>There is a <i>q</i>-analog to this theorem called the <i>q</i>-Vandermonde identity.
<br/>Vandermonde's identity can be generalized in numerous ways, including to the identity
<br/>
<br/><dl><dd><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 {n_{1}+\\\\dots +n_{p} \\\\choose m}=\\\\sum _{k_{1}+\\\\cdots +k_{p}=m}{n_{1} \\\\choose k_{1}}{n_{2} \\\\choose k_{2}}\\\\cdots {n_{p} \\\\choose k_{p}}.}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mrow>
<br/>                <msub>
<br/>                  <mi>n</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mn>1</mn>
<br/>                  </mrow>
<br/>                </msub>
<br/>                <mo>+</mo>
<br/>                <mo>⋯<!-- ⋯ --></mo>
<br/>                <mo>+</mo>
<br/>                <msub>
<br/>                  <mi>n</mi>
<br/>                  <mrow class="MJX-TeXAtom-ORD">
<br/>                    <mi>p</mi>
<br/>                  </mrow>
<br/>                </msub>
<br/>              </mrow>
<br/>              <mi>m</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <munder>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <msub>
<br/>              <mi>k</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mn>1</mn>
<br/>              </mrow>
<br/>            </msub>
<br/>            <mo>+</mo>
<br/>            <mo>⋯<!-- ⋯ --></mo>
<br/>            <mo>+</mo>
<br/>            <msub>
<br/>              <mi>k</mi>
<br/>              <mrow class="MJX-TeXAtom-ORD">
<br/>                <mi>p</mi>
<br/>              </mrow>
<br/>            </msub>
<br/>            <mo>=</mo>
<br/>            <mi>m</mi>
<br/>          </mrow>
<br/>        </munder>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <msub>
<br/>                <mi>n</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>1</mn>
<br/>                </mrow>
<br/>              </msub>
<br/>              <msub>
<br/>                <mi>k</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>1</mn>
<br/>                </mrow>
<br/>              </msub>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <msub>
<br/>                <mi>n</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msub>
<br/>              <msub>
<br/>                <mi>k</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mn>2</mn>
<br/>                </mrow>
<br/>              </msub>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>⋯<!-- ⋯ --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <msub>
<br/>                <mi>n</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>p</mi>
<br/>                </mrow>
<br/>              </msub>
<br/>              <msub>
<br/>                <mi>k</mi>
<br/>                <mrow class="MJX-TeXAtom-ORD">
<br/>                  <mi>p</mi>
<br/>                </mrow>
<br/>              </msub>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>.</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {n_{1}+\\\\dots +n_{p} \\\\choose m}=\\\\sum _{k_{1}+\\\\cdots +k_{p}=m}{n_{1} \\\\choose k_{1}}{n_{2} \\\\choose k_{2}}\\\\cdots {n_{p} \\\\choose k_{p}}.}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6430edc1272e716e3f29a78ba10d6374da6a2d0" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -3.505ex; width:52.557ex; height:7.176ex;" alt="{\\\\displaystyle {n_{1}+\\\\dots +n_{p} \\\\choose m}=\\\\sum _{k_{1}+\\\\cdots +k_{p}=m}{n_{1} \\\\choose k_{1}}{n_{2} \\\\choose k_{2}}\\\\cdots {n_{p} \\\\choose k_{p}}.}"> 
<br/>(Wikipedia, The Free Encyclopedia, <a href="https://en.wikipedia.org/wiki/Vandermonde%27s_identity">https://en.wikipedia.org/wiki/Vandermonde%27s_identity</a>)"""@en, """En mathématiques combinatoires, l'<b>identité de Vandermonde</b>, ainsi nommée en l'honneur d'Alexandre-Théophile Vandermonde (1772), ou <b>formule de convolution</b>, affirme que, pour des entiers naturels <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 k,m,n}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>k</mi>
<br/>        <mo>,</mo>
<br/>        <mi>m</mi>
<br/>        <mo>,</mo>
<br/>        <mi>n</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle k,m,n}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc55a7f1794c8e4f6136fded55728fe2bb1e01c2" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:6.714ex; height:2.509ex;" alt="{\\\\displaystyle k,m,n}"></span>, on a
<br/>
<br/><dl><dd><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 {\\inom {m+n}{k}}=\\\\sum _{\\\\overset {0\\\\leqslant i,j\\\\leqslant k}{i+j=k}}{\\inom {m}{i}}{\\inom {n}{j}}=\\\\sum _{i=0}^{k}{\\inom {m}{i}}{\\inom {n}{k-i}}=\\\\sum _{j=0}^{k}{\\inom {m}{k-j}}{\\inom {n}{j}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mrow>
<br/>                <mi>m</mi>
<br/>                <mo>+</mo>
<br/>                <mi>n</mi>
<br/>              </mrow>
<br/>              <mi>k</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <munder>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mover>
<br/>              <mrow>
<br/>                <mi>i</mi>
<br/>                <mo>+</mo>
<br/>                <mi>j</mi>
<br/>                <mo>=</mo>
<br/>                <mi>k</mi>
<br/>              </mrow>
<br/>              <mrow>
<br/>                <mn>0</mn>
<br/>                <mo>⩽<!-- ⩽ --></mo>
<br/>                <mi>i</mi>
<br/>                <mo>,</mo>
<br/>                <mi>j</mi>
<br/>                <mo>⩽<!-- ⩽ --></mo>
<br/>                <mi>k</mi>
<br/>              </mrow>
<br/>            </mover>
<br/>          </mrow>
<br/>        </munder>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mi>m</mi>
<br/>              <mi>i</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mi>n</mi>
<br/>              <mi>j</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>i</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mi>m</mi>
<br/>              <mi>i</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mi>n</mi>
<br/>              <mrow>
<br/>                <mi>k</mi>
<br/>                <mo>−<!-- − --></mo>
<br/>                <mi>i</mi>
<br/>              </mrow>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <munderover>
<br/>          <mo>∑<!-- ∑ --></mo>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>j</mi>
<br/>            <mo>=</mo>
<br/>            <mn>0</mn>
<br/>          </mrow>
<br/>          <mrow class="MJX-TeXAtom-ORD">
<br/>            <mi>k</mi>
<br/>          </mrow>
<br/>        </munderover>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mi>m</mi>
<br/>              <mrow>
<br/>                <mi>k</mi>
<br/>                <mo>−<!-- − --></mo>
<br/>                <mi>j</mi>
<br/>              </mrow>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mi>n</mi>
<br/>              <mi>j</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {\\inom {m+n}{k}}=\\\\sum _{\\\\overset {0\\\\leqslant i,j\\\\leqslant k}{i+j=k}}{\\inom {m}{i}}{\\inom {n}{j}}=\\\\sum _{i=0}^{k}{\\inom {m}{i}}{\\inom {n}{k-i}}=\\\\sum _{j=0}^{k}{\\inom {m}{k-j}}{\\inom {n}{j}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/340b26c826c0fac09d4c6af4844fe69d8d620479" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -5.005ex; width:68.945ex; height:9.343ex;" alt="{\\\\displaystyle {\\inom {m+n}{k}}=\\\\sum _{\\\\overset {0\\\\leqslant i,j\\\\leqslant k}{i+j=k}}{\\inom {m}{i}}{\\inom {n}{j}}=\\\\sum _{i=0}^{k}{\\inom {m}{i}}{\\inom {n}{k-i}}=\\\\sum _{j=0}^{k}{\\inom {m}{k-j}}{\\inom {n}{j}}}"></span></dd></dl>
<br/>où les nombres <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 {b \\\\choose a}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mi>b</mi>
<br/>              <mi>a</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {b \\\\choose a}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92ee3f7f7a303922e934d497a9513fc56c7a521" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:4.651ex; height:6.176ex;" alt="{\\\\displaystyle {b \\\\choose a}}"></span>, avec <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 a,b\\\\in \\\\mathbb {N} }">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>a</mi>
<br/>        <mo>,</mo>
<br/>        <mi>b</mi>
<br/>        <mo>∈<!-- ∈ --></mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mi mathvariant="double-struck">N</mi>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle a,b\\\\in \\\\mathbb {N} }</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b604733da4cfdd0d8c6c6956d592b26c2e1fc351" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.671ex; width:7.78ex; height:2.509ex;" alt="{\\\\displaystyle a,b\\\\in \\\\mathbb {N} }"></span>, sont des coefficients binomiaux, c'est-à-dire que <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 {b \\\\choose a}={\\rac {b!}{a!\\\\,(b-a)!}}}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mi>b</mi>
<br/>              <mi>a</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mfrac>
<br/>            <mrow>
<br/>              <mi>b</mi>
<br/>              <mo>!</mo>
<br/>            </mrow>
<br/>            <mrow>
<br/>              <mi>a</mi>
<br/>              <mo>!</mo>
<br/>              <mspace width="thinmathspace"></mspace>
<br/>              <mo stretchy="false">(</mo>
<br/>              <mi>b</mi>
<br/>              <mo>−<!-- − --></mo>
<br/>              <mi>a</mi>
<br/>              <mo stretchy="false">)</mo>
<br/>              <mo>!</mo>
<br/>            </mrow>
<br/>          </mfrac>
<br/>        </mrow>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {b \\\\choose a}={\\rac {b!}{a!\\\\,(b-a)!}}}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4a91ee1b7c80d172f568047578b0d50abb7e749" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.671ex; width:18.373ex; height:6.343ex;" alt="{\\\\displaystyle {b \\\\choose a}={\\rac {b!}{a!\\\\,(b-a)!}}}"></span> si <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 0\\\\leqslant a\\\\leqslant b}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mn>0</mn>
<br/>        <mo>⩽<!-- ⩽ --></mo>
<br/>        <mi>a</mi>
<br/>        <mo>⩽<!-- ⩽ --></mo>
<br/>        <mi>b</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle 0\\\\leqslant a\\\\leqslant b}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ef8081f51a9c7c8c85a5bb0ea58f0952071544bb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.505ex; width:9.587ex; height:2.343ex;" alt="{\\\\displaystyle 0\\\\leqslant a\\\\leqslant b}"></span> (le point d'exclamation «&nbsp;!&nbsp;» désignant la factorielle) et <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 {b \\\\choose a}=0}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mrow class="MJX-TeXAtom-ORD">
<br/>          <mrow>
<br/>            <mrow class="MJX-TeXAtom-OPEN">
<br/>              <mo maxsize="2.047em" minsize="2.047em">(</mo>
<br/>            </mrow>
<br/>            <mfrac linethickness="0">
<br/>              <mi>b</mi>
<br/>              <mi>a</mi>
<br/>            </mfrac>
<br/>            <mrow class="MJX-TeXAtom-CLOSE">
<br/>              <mo maxsize="2.047em" minsize="2.047em">)</mo>
<br/>            </mrow>
<br/>          </mrow>
<br/>        </mrow>
<br/>        <mo>=</mo>
<br/>        <mn>0</mn>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle {b \\\\choose a}=0}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0cee2c7b187e55b92cfa61135dc62f278bfa28f" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -2.505ex; width:8.912ex; height:6.176ex;" alt="{\\\\displaystyle {b \\\\choose a}=0}"></span> si <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 a>b}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mi>a</mi>
<br/>        <mo>&gt;</mo>
<br/>        <mi>b</mi>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle a&gt;b}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83fc0063781fb9bf4ec7608b2fd11ed6d5b05a13" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.338ex; width:5.326ex; height:2.176ex;" alt="a>b"></span>.
<br/>Les contributions non nulles à la somme de droite proviennent des valeurs de <span class="texhtml mvar" style="font-style:italic;">j</span> pour lesquelles les coefficients binomiaux sont non nuls, c'est-à-dire pour <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 \\\\max(0,k-m)\\\\leqslant j\\\\leqslant \\\\min(n,k)}">
<br/>  <semantics>
<br/>    <mrow class="MJX-TeXAtom-ORD">
<br/>      <mstyle displaystyle="true" scriptlevel="0">
<br/>        <mo movablelimits="true" form="prefix">max</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mn>0</mn>
<br/>        <mo>,</mo>
<br/>        <mi>k</mi>
<br/>        <mo>−<!-- − --></mo>
<br/>        <mi>m</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>        <mo>⩽<!-- ⩽ --></mo>
<br/>        <mi>j</mi>
<br/>        <mo>⩽<!-- ⩽ --></mo>
<br/>        <mo movablelimits="true" form="prefix">min</mo>
<br/>        <mo stretchy="false">(</mo>
<br/>        <mi>n</mi>
<br/>        <mo>,</mo>
<br/>        <mi>k</mi>
<br/>        <mo stretchy="false">)</mo>
<br/>      </mstyle>
<br/>    </mrow>
<br/>    <annotation encoding="application/x-tex">{\\\\displaystyle \\\\max(0,k-m)\\\\leqslant j\\\\leqslant \\\\min(n,k)}</annotation>
<br/>  </semantics>
<br/></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce21c09967aa56986b5c1c2fd257bfabfdfa2d0a" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:30.903ex; height:2.843ex;" alt="{\\\\displaystyle \\\\max(0,k-m)\\\\leqslant j\\\\leqslant \\\\min(n,k)}"></span>. 
<br/>(Wikipedia, L'Encylopédie Libre, <a href="https://fr.wikipedia.org/wiki/Identit%C3%A9_de_Vandermonde">https://fr.wikipedia.org/wiki/Identit%C3%A9_de_Vandermonde</a>)"""@fr ;
  dc:created "2023-07-13"^^xsd:date ;
  dc:modified "2023-07-13"^^xsd:date .