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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Pyramide de Pascal</span></h1>
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<p>En <a href="Math%C3%A9matiques" title="Mathématiques">mathématiques</a>, la <b>pyramide de Pascal</b> (ou <b>tétraèdre de Pascal</b>) est une généralisation tridimensionnelle du <a href="Triangle_de_Pascal" title="Triangle de Pascal">triangle de Pascal</a>. De même que le triangle de <a href="Blaise_Pascal" title="Blaise Pascal">Pascal</a> donne les <a href="Coefficients_binomiaux" class="mw-redirect" title="Coefficients binomiaux">coefficients binomiaux</a>, la pyramide de Pascal donne les <a href="Formule_du_trin%C3%B4me_de_Newton" title="Formule du trinôme de Newton">coefficients trinomiaux</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Coefficients_trinomiaux">Coefficients trinomiaux</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Présentation_et_utilité"><span id="Pr.C3.A9sentation_et_utilit.C3.A9"></span>Présentation et utilité</h3></div>
<p>Les coefficients trinomiaux constituent un cas particulier des <a href="Formule_du_multin%C3%B4me_de_Newton" title="Formule du multinôme de Newton">coefficients multinomiaux</a> ; ils s'écrivent sous la forme <span class="mwe-math-element mwe-math-element-inline"><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 \choose i,j,k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mi>n</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {n \choose i,j,k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f38e9592f9b120963842e11a7949d592844eb08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.461ex; height:6.176ex;" alt="{\displaystyle {n \choose i,j,k}}" loading="lazy"></span> où <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j,k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i,j,k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9550e66fe7e601c4f58bbc9c19ba226301149cde.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.04ex; height:2.509ex;" alt="{\displaystyle i,j,k}" loading="lazy"></span> sont trois <a href="Entier_naturel" title="Entier naturel">entiers naturels</a> (positifs ou nuls) et <span class="mwe-math-element mwe-math-element-inline"><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=i+j+k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mi>i</mi>
<mo>+</mo>
<mi>j</mi>
<mo>+</mo>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=i+j+k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe75790be3ea51cd12ae24e64457f5e31910891f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.146ex; height:2.509ex;" alt="{\displaystyle n=i+j+k}" loading="lazy"></span>. Ils sont définis par la formule <span class="mwe-math-element mwe-math-element-inline"><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 \choose i,j,k}={\frac {n!}{i!j!k!}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mi>n</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>n</mi>
<mo>!</mo>
</mrow>
<mrow>
<mi>i</mi>
<mo>!</mo>
<mi>j</mi>
<mo>!</mo>
<mi>k</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {n \choose i,j,k}={\frac {n!}{i!j!k!}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47ea69ca1ebb1d8a68282b5c8ce1d1fbeff245bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:17.308ex; height:6.176ex;" alt="{\displaystyle {n \choose i,j,k}={\frac {n!}{i!j!k!}}}" loading="lazy"></span>. Comme tous les coefficients multinomiaux, leur intérêt est multiple ; on peut les retrouver dans les cas suivants :
</p>
<ul><li>En <a href="Alg%C3%A8bre" title="Algèbre">algèbre</a> avec le développement du <a href="Formule_du_trin%C3%B4me_de_Newton" title="Formule du trinôme de Newton">trinôme de Newton</a>.</li>
<li>En <a href="D%C3%A9nombrement" title="Dénombrement">dénombrement</a>, <span class="mwe-math-element mwe-math-element-inline"><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 \choose i,j,k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mi>n</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {n \choose i,j,k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f38e9592f9b120963842e11a7949d592844eb08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.461ex; height:6.176ex;" alt="{\displaystyle {n \choose i,j,k}}" loading="lazy"></span> est le nombre d'arrangements possibles d'une population de <span class="texhtml mvar" style="font-style:italic;">n</span> objets composée de <span class="texhtml mvar" style="font-style:italic;">i</span> objets d'une espèce A, <span class="texhtml mvar" style="font-style:italic;">j</span> objets d'une espèce B et <span class="texhtml mvar" style="font-style:italic;">k</span> objets d'une espèce C, les objets d'une même espèce étant indiscernables et leurs positions relatives n'important donc pas.</li>
<li>En <a href="Statistique" title="Statistique">statistique</a>, du fait de la propriété précédente.</li></ul>
<div class="mw-heading mw-heading3"><h3 id="Lien_entre_les_coefficients_trinomiaux_et_la_pyramide_de_Pascal">Lien entre les coefficients trinomiaux et la pyramide de Pascal</h3></div>
<p>La pyramide de Pascal se construit grâce à la loi de récurrence (de type <a href="Triangle_de_Pascal" title="Triangle de Pascal">relation de Pascal</a>) concernant les coefficients trinomiaux suivante : <span class="mwe-math-element mwe-math-element-inline"><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 \choose i,j,k}={n-1 \choose i-1,j,k}+{n-1 \choose i,j-1,k}+{n-1 \choose i,j,k-1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mi>n</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>i</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {n \choose i,j,k}={n-1 \choose i-1,j,k}+{n-1 \choose i,j-1,k}+{n-1 \choose i,j,k-1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/13043ec03dfb900f00ddb0357c4d34d3efb37057.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:54.631ex; height:6.176ex;" alt="{\displaystyle {n \choose i,j,k}={n-1 \choose i-1,j,k}+{n-1 \choose i,j-1,k}+{n-1 \choose i,j,k-1}}" loading="lazy"></span>, vraie pour tous les triplets <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j,k)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (i,j,k)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca15aed635d1550057e6ea4cc1a5d125d225706f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.849ex; height:2.843ex;" alt="{\displaystyle (i,j,k)}" loading="lazy"></span> d'entiers naturel avec <span class="mwe-math-element mwe-math-element-inline"><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=i+j+k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mi>i</mi>
<mo>+</mo>
<mi>j</mi>
<mo>+</mo>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=i+j+k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe75790be3ea51cd12ae24e64457f5e31910891f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.146ex; height:2.509ex;" alt="{\displaystyle n=i+j+k}" loading="lazy"></span>.
</p><p>Cette règle reste vraie dans les cas où <span class="texhtml mvar" style="font-style:italic;">i, j</span> ou <span class="texhtml mvar" style="font-style:italic;">k</span> sont égaux à 0, à condition de prendre la convention suivante: <span class="mwe-math-element mwe-math-element-inline"><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' \choose i',j',k'}=0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<msup>
<mi>n</mi>
<mo>′</mo>
</msup>
<mrow>
<msup>
<mi>i</mi>
<mo>′</mo>
</msup>
<mo>,</mo>
<msup>
<mi>j</mi>
<mo>′</mo>
</msup>
<mo>,</mo>
<msup>
<mi>k</mi>
<mo>′</mo>
</msup>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>=</mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {n' \choose i',j',k'}=0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bac1f7c366fbf25fc91a943a9241fdfb09dd7de2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.776ex; height:6.176ex;" alt="{\displaystyle {n' \choose i',j',k'}=0}" loading="lazy"></span> si <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i'\leqslant -1{\text{ ou }}j'\leqslant -1{\text{ ou }}k'\leqslant -1{\text{ et }}n'=i'+j'+k'\geqslant 0}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mi>i</mi>
<mo>′</mo>
</msup>
<mo>⩽<!-- ⩽ --></mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mtext> ou </mtext>
</mrow>
<msup>
<mi>j</mi>
<mo>′</mo>
</msup>
<mo>⩽<!-- ⩽ --></mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mtext> ou </mtext>
</mrow>
<msup>
<mi>k</mi>
<mo>′</mo>
</msup>
<mo>⩽<!-- ⩽ --></mo>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mrow class="MJX-TeXAtom-ORD">
<mtext> et </mtext>
</mrow>
<msup>
<mi>n</mi>
<mo>′</mo>
</msup>
<mo>=</mo>
<msup>
<mi>i</mi>
<mo>′</mo>
</msup>
<mo>+</mo>
<msup>
<mi>j</mi>
<mo>′</mo>
</msup>
<mo>+</mo>
<msup>
<mi>k</mi>
<mo>′</mo>
</msup>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i'\leqslant -1{\text{ ou }}j'\leqslant -1{\text{ ou }}k'\leqslant -1{\text{ et }}n'=i'+j'+k'\geqslant 0}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/03b130870f34dfd25fd94c2646ba369c5b990f0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:53.709ex; height:2.843ex;" alt="{\displaystyle i'\leqslant -1{\text{ ou }}j'\leqslant -1{\text{ ou }}k'\leqslant -1{\text{ et }}n'=i'+j'+k'\geqslant 0}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading4"><h4 id="Démonstration"><span id="D.C3.A9monstration"></span>Démonstration</h4></div>
<p>La démonstration consiste à rechercher une factorisation de l'expression
</p><p><span class="mwe-math-element mwe-math-element-inline"><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 \choose i-1,j,k}+{n-1 \choose i,j-1,k}+{n-1 \choose i,j,k-1}={\frac {(n-1)!}{(i-1)!j!k!}}+{\frac {(n-1)!}{i!(j-1)!k!}}+{\frac {(n-1)!}{i!j!(k-1)!}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>i</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
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</mrow>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
<mrow>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
<mi>j</mi>
<mo>!</mo>
<mi>k</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
<mrow>
<mi>i</mi>
<mo>!</mo>
<mo stretchy="false">(</mo>
<mi>j</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
<mi>k</mi>
<mo>!</mo>
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</mfrac>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
<mrow>
<mi>i</mi>
<mo>!</mo>
<mi>j</mi>
<mo>!</mo>
<mo stretchy="false">(</mo>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
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</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {n-1 \choose i-1,j,k}+{n-1 \choose i,j-1,k}+{n-1 \choose i,j,k-1}={\frac {(n-1)!}{(i-1)!j!k!}}+{\frac {(n-1)!}{i!(j-1)!k!}}+{\frac {(n-1)!}{i!j!(k-1)!}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9a7345c7d228f774b3f18de5fe3f5712acc513c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:86.533ex; height:6.509ex;" alt="{\displaystyle {n-1 \choose i-1,j,k}+{n-1 \choose i,j-1,k}+{n-1 \choose i,j,k-1}={\frac {(n-1)!}{(i-1)!j!k!}}+{\frac {(n-1)!}{i!(j-1)!k!}}+{\frac {(n-1)!}{i!j!(k-1)!}}}" loading="lazy"></span>
</p><p>On peut pour cela commencer par mettre le terme (<span class="texhtml mvar" style="font-style:italic;">n</span> - 1)! en facteur, puis en cherchant à réduire les trois termes au même dénominateur on va remarquer que <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{(i-1)!}}={\frac {i}{i!}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mn>1</mn>
<mrow>
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mi>i</mi>
<mrow>
<mi>i</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{(i-1)!}}={\frac {i}{i!}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/968bfa69ff0bf42cb3722c5aebc041e2eb9eef4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.481ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{(i-1)!}}={\frac {i}{i!}}}" loading="lazy"></span> , un dénominateur commun est donc <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i!j!k!}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>!</mo>
<mi>j</mi>
<mo>!</mo>
<mi>k</mi>
<mo>!</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i!j!k!}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/08a31d3da96003d7420b5a9be6592b070cdbf24c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.912ex; height:2.509ex;" alt="{\displaystyle i!j!k!}" loading="lazy"></span> .
</p><p>On va donc avoir : <span class="mwe-math-element mwe-math-element-inline"><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 \choose i-1,j,k}+{n-1 \choose i,j-1,k}+{n-1 \choose i,j,k-1}={\frac {(n-1)!}{i!j!k!}}\left(i+j+k\right)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>i</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
</mrow>
<mrow>
<mi>i</mi>
<mo>!</mo>
<mi>j</mi>
<mo>!</mo>
<mi>k</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mrow>
<mo>(</mo>
<mrow>
<mi>i</mi>
<mo>+</mo>
<mi>j</mi>
<mo>+</mo>
<mi>k</mi>
</mrow>
<mo>)</mo>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {n-1 \choose i-1,j,k}+{n-1 \choose i,j-1,k}+{n-1 \choose i,j,k-1}={\frac {(n-1)!}{i!j!k!}}\left(i+j+k\right)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2daca454f6333c39845e6073e6aa9168e934b1d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:65.709ex; height:6.343ex;" alt="{\displaystyle {n-1 \choose i-1,j,k}+{n-1 \choose i,j-1,k}+{n-1 \choose i,j,k-1}={\frac {(n-1)!}{i!j!k!}}\left(i+j+k\right)}" loading="lazy"></span>
</p><p>Or <span class="mwe-math-element mwe-math-element-inline"><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=i+j+k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mi>i</mi>
<mo>+</mo>
<mi>j</mi>
<mo>+</mo>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=i+j+k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe75790be3ea51cd12ae24e64457f5e31910891f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.146ex; height:2.509ex;" alt="{\displaystyle n=i+j+k}" loading="lazy"></span> et <span class="mwe-math-element mwe-math-element-inline"><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)!n=n!}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo stretchy="false">)</mo>
<mo>!</mo>
<mi>n</mi>
<mo>=</mo>
<mi>n</mi>
<mo>!</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (n-1)!n=n!}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c302c998dabe620388344cb07fe51cbe490384cc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.388ex; height:2.843ex;" alt="{\displaystyle (n-1)!n=n!}" loading="lazy"></span> donc:
</p><p><span class="mwe-math-element mwe-math-element-inline"><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 \choose i-1,j,k}+{n-1 \choose i,j-1,k}+{n-1 \choose i,j,k-1}={\frac {n!}{i!j!k!}}={n \choose i,j,k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
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<mrow>
<mi>i</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mi>j</mi>
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<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>+</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
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<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>n</mi>
<mo>!</mo>
</mrow>
<mrow>
<mi>i</mi>
<mo>!</mo>
<mi>j</mi>
<mo>!</mo>
<mi>k</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mi>n</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {n-1 \choose i-1,j,k}+{n-1 \choose i,j-1,k}+{n-1 \choose i,j,k-1}={\frac {n!}{i!j!k!}}={n \choose i,j,k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9cb52a3a1d931a8ce934223816aff9d890c46ce3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:63.478ex; height:6.176ex;" alt="{\displaystyle {n-1 \choose i-1,j,k}+{n-1 \choose i,j-1,k}+{n-1 \choose i,j,k-1}={\frac {n!}{i!j!k!}}={n \choose i,j,k}}" loading="lazy"></span>
</p>
<div class="mw-heading mw-heading2"><h2 id="Lecture_de_la_pyramide_de_Pascal">Lecture de la pyramide de Pascal</h2></div>
<p>La pyramide est construite étage par étage en commençant par le sommet (<span class="texhtml mvar" style="font-style:italic;">n</span> = 0) et en descendant (incrémentation de <span class="texhtml mvar" style="font-style:italic;">n</span>). Les premiers étages du haut sont ainsi constitués :
</p>
<pre>1 1 1 1 1
1 1 2 2 3 3 4 4
1 2 1 3 6 3 6 12 6
1 3 3 1 4 12 12 4
1 4 6 4 1
</pre>
<p>Du fait qu'à l'étage <span class="texhtml mvar" style="font-style:italic;">n</span>, les entiers naturel <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (i,j,k)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (i,j,k)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ca15aed635d1550057e6ea4cc1a5d125d225706f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.849ex; height:2.843ex;" alt="{\displaystyle (i,j,k)}" loading="lazy"></span> doivent respecter la relation <span class="mwe-math-element mwe-math-element-inline"><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=i+j+k}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>n</mi>
<mo>=</mo>
<mi>i</mi>
<mo>+</mo>
<mi>j</mi>
<mo>+</mo>
<mi>k</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle n=i+j+k}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe75790be3ea51cd12ae24e64457f5e31910891f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.146ex; height:2.509ex;" alt="{\displaystyle n=i+j+k}" loading="lazy"></span>, on repère la position du coefficient trinomial <span class="mwe-math-element mwe-math-element-inline"><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 \choose i,j,k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mi>n</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {n \choose i,j,k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8f38e9592f9b120963842e11a7949d592844eb08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.461ex; height:6.176ex;" alt="{\displaystyle {n \choose i,j,k}}" loading="lazy"></span> sur l'étage <span class="texhtml mvar" style="font-style:italic;">n</span> à l'aide de conventions telles que celles-ci (appliquée dans ce cas à <span class="texhtml mvar" style="font-style:italic;">n</span> = 3):
</p>
<pre> k=0
\
k=1 1 - j=3
\ / \
k=2 3 - 3 - j=2
\ / \ / \
k=3 3 - 6 - 3 - j=1
\ / \ / \ / \
1 - 3 - 3 - 1 - j=0
/ / / /
i=0 i=1 i=2 i=3
</pre>
<p>Ainsi, avec les conventions adoptées, on peut repérer que <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {3 \choose 1,1,1}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mn>3</mn>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
<mo>,</mo>
<mn>1</mn>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {3 \choose 1,1,1}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e26a128a3552a5e99fe3506f3411e757a022954e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.976ex; height:6.176ex;" alt="{\displaystyle {3 \choose 1,1,1}}" loading="lazy"></span> est le deuxième coefficient (<span class="texhtml mvar" style="font-style:italic;">i</span> = 1) présent sur la deuxième rangée (<span class="texhtml mvar" style="font-style:italic;">j</span> = 1) en partant du bas et est donc égal à 6. De même <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {3 \choose 1,2,0}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mn>3</mn>
<mrow>
<mn>1</mn>
<mo>,</mo>
<mn>2</mn>
<mo>,</mo>
<mn>0</mn>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {3 \choose 1,2,0}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bd6d648b935fba211f2be8d87fcbc8d029d4f017.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.976ex; height:6.176ex;" alt="{\displaystyle {3 \choose 1,2,0}}" loading="lazy"></span> est le deuxième (<span class="texhtml mvar" style="font-style:italic;">i</span> = 1) élément de la troisième (<span class="texhtml mvar" style="font-style:italic;">j</span> = 2) rangée en partant du bas, il est donc égal à 3, on remarquera que l'élément en question se trouve sur la rangée d'indice <span class="texhtml mvar" style="font-style:italic;">k</span> = 0.
</p><p>Le lien avec la relation de Pascal se retrouve du fait que la valeur d'un élément est la somme des trois (deux si on se trouve sur une face, 1 sur une arête) éléments se trouvant directement au-dessus de celui-ci dans l'étage précédent. Par exemple, pour l'étage <span class="texhtml mvar" style="font-style:italic;">n</span> = 3, le 6 central se trouve être la somme des trois 2 se trouvant juste au-dessus dans l'étage <span class="texhtml mvar" style="font-style:italic;">n</span> = 2.
</p>
<div class="mw-heading mw-heading2"><h2 id="Propriétés"><span id="Propri.C3.A9t.C3.A9s"></span>Propriétés</h2></div>
<p>Chaque face de la pyramide de Pascal est identifiable à un triangle de Pascal. Cela peut être expliqué soit par récurrence, mais il est plus simple de constater qu'une face de la pyramide correspond à un plan vérifiant <span class="texhtml mvar" style="font-style:italic;">i</span> = 0, <span class="texhtml mvar" style="font-style:italic;">j</span> = 0 ou <span class="texhtml mvar" style="font-style:italic;">k</span> = 0, et puisque <span class="mwe-math-element mwe-math-element-inline"><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!=1}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mn>0</mn>
<mo>!</mo>
<mo>=</mo>
<mn>1</mn>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 0!=1}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/22956a0fa255c6c9562eab440f8c23c2954a6cf4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.07ex; height:2.176ex;" alt="{\displaystyle 0!=1}" loading="lazy"></span>, on a <span class="mwe-math-element mwe-math-element-inline"><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 \choose 0,j,k}={\frac {n!}{j!k!}}={n \choose j,k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mi>n</mi>
<mrow>
<mn>0</mn>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi>n</mi>
<mo>!</mo>
</mrow>
<mrow>
<mi>j</mi>
<mo>!</mo>
<mi>k</mi>
<mo>!</mo>
</mrow>
</mfrac>
</mrow>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mi>n</mi>
<mrow>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {n \choose 0,j,k}={\frac {n!}{j!k!}}={n \choose j,k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84820ccf3e221188bffb251cee1300e80fdeefec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.941ex; height:6.176ex;" alt="{\displaystyle {n \choose 0,j,k}={\frac {n!}{j!k!}}={n \choose j,k}}" loading="lazy"></span> que l'on peut aussi noter <span class="mwe-math-element mwe-math-element-inline"><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 \choose k}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mi>n</mi>
<mi>k</mi>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {n \choose k}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8cc51538192fdf193790d4378c3a998a6b94262.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:4.816ex; height:6.176ex;" alt="{\displaystyle {n \choose k}}" loading="lazy"></span> ou <span class="mwe-math-element mwe-math-element-inline"><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 \choose j}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mi>n</mi>
<mi>j</mi>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {n \choose j}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9aa4c74b27e886f8c82a6b9e011da97aa8afc945.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:4.816ex; height:6.176ex;" alt="{\displaystyle {n \choose j}}" loading="lazy"></span>.
</p><p>La pyramide de Pascal peut être utilisée dans le développement du trinôme du newton du fait de la formule :
</p><p><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x+y+z)^{n}=\sum _{i,j,k\geqslant 0,i+j+k=n}{{n \choose i,j,k}x^{i}y^{j}z^{k}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>+</mo>
<mi>y</mi>
<mo>+</mo>
<mi>z</mi>
<msup>
<mo stretchy="false">)</mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
<mo>=</mo>
<munder>
<mo>∑<!-- ∑ --></mo>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
<mo>⩾<!-- ⩾ --></mo>
<mn>0</mn>
<mo>,</mo>
<mi>i</mi>
<mo>+</mo>
<mi>j</mi>
<mo>+</mo>
<mi>k</mi>
<mo>=</mo>
<mi>n</mi>
</mrow>
</munder>
<mrow class="MJX-TeXAtom-ORD">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mi>n</mi>
<mrow>
<mi>i</mi>
<mo>,</mo>
<mi>j</mi>
<mo>,</mo>
<mi>k</mi>
</mrow>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
<msup>
<mi>x</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>i</mi>
</mrow>
</msup>
<msup>
<mi>y</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>j</mi>
</mrow>
</msup>
<msup>
<mi>z</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>k</mi>
</mrow>
</msup>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle (x+y+z)^{n}=\sum _{i,j,k\geqslant 0,i+j+k=n}{{n \choose i,j,k}x^{i}y^{j}z^{k}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e5c23adaa71dc0669bc81f5c18c41bea9ca09994.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:43.104ex; height:7.009ex;" alt="{\displaystyle (x+y+z)^{n}=\sum _{i,j,k\geqslant 0,i+j+k=n}{{n \choose i,j,k}x^{i}y^{j}z^{k}}}" loading="lazy"></span>
</p><p>On peut déduire de cette formule ou par récurrence que la somme des éléments de l'étage <span class="texhtml mvar" style="font-style:italic;">n</span> est égale à <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3^{n}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<msup>
<mn>3</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
</mrow>
</msup>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle 3^{n}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/193abd21d79ceb2992929ab3b3a1ee97d2afb6a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.381ex; height:2.343ex;" alt="{\displaystyle 3^{n}}" loading="lazy"></span>.
</p><p>Une pyramide de Pascal allant de l'étage 0 à l'étage <span class="texhtml mvar" style="font-style:italic;">n</span> inclus est constituée de <span class="mwe-math-element mwe-math-element-inline"><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+3}{3}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>+</mo>
<mn>3</mn>
</mrow>
<mn>3</mn>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\binom {n+3}{3}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7b423ccbd59eefaf5e9b21a7ee99c7a427cf3165.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.819ex; height:6.176ex;" alt="{\displaystyle {\binom {n+3}{3}}}" loading="lazy"></span> nombres, et la somme de ces nombres est égale à <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {3^{n+1}-1}{2}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mn>3</mn>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mn>2</mn>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {3^{n+1}-1}{2}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b9f855ab576f5f81310fd7506ed6b09af9c707d3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.32ex; height:5.676ex;" alt="{\displaystyle {\frac {3^{n+1}-1}{2}}}" loading="lazy"></span>.
</p><p>Plus généralement, un <a href="Simplexe" title="Simplexe">simplexe</a> de Pascal de dimension <span class="texhtml mvar" style="font-style:italic;">d</span> (formé des <a href="Formule_du_multin%C3%B4me_de_Newton" title="Formule du multinôme de Newton">coefficients <i>d</i>-nomiaux</a>) allant de l'étage 0 à l'étage <span class="texhtml mvar" style="font-style:italic;">n</span> inclus est constitué de <span class="mwe-math-element mwe-math-element-inline"><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+d}{d}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mrow>
<mrow class="MJX-TeXAtom-OPEN">
<mo maxsize="2.047em" minsize="2.047em">(</mo>
</mrow>
<mfrac linethickness="0">
<mrow>
<mi>n</mi>
<mo>+</mo>
<mi>d</mi>
</mrow>
<mi>d</mi>
</mfrac>
<mrow class="MJX-TeXAtom-CLOSE">
<mo maxsize="2.047em" minsize="2.047em">)</mo>
</mrow>
</mrow>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\binom {n+d}{d}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1bf56c7f07120a0387b8527602b15689a7723715.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:8.872ex; height:6.176ex;" alt="{\displaystyle {\binom {n+d}{d}}}" loading="lazy"></span> nombres, et la somme de ces nombres est égale à <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{n+1}-1}{d-1}}}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<msup>
<mi>d</mi>
<mrow class="MJX-TeXAtom-ORD">
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
</mrow>
</msup>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
<mrow>
<mi>d</mi>
<mo>−<!-- − --></mo>
<mn>1</mn>
</mrow>
</mfrac>
</mrow>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle {\frac {d^{n+1}-1}{d-1}}}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/90213b698de92d0dd243ee53c6a9c0ed03f64109.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:9.376ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{n+1}-1}{d-1}}}" loading="lazy"></span>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Notes_et_références"><span id="Notes_et_r.C3.A9f.C3.A9rences"></span>Notes et références</h2></div>
<div style="font-size:85%; padding-left:1.6em; margin:0.3em 0;"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> Cet article est partiellement ou en totalité issu de l’article de Wikipédia en anglais intitulé <span class="">« <a class="external text" href="https://en.wikipedia.org/wiki/Pascal%27s_pyramid?oldid=6535194">Pascal's pyramid</a> » <small>(<a class="external text" href="https://en.wikipedia.org/wiki/Pascal%27s_pyramid?action=history">voir la liste des auteurs</a>)</small></span>.</div>
<div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2></div>
<div class="mw-heading mw-heading3"><h3 id="Articles_connexes">Articles connexes</h3></div>
<ul><li><a href="Formule_du_bin%C3%B4me_de_Newton" title="Formule du binôme de Newton">Formule du binôme de Newton</a></li>
<li><a href="Formule_du_trin%C3%B4me_de_Newton" title="Formule du trinôme de Newton">Formule du trinôme de Newton</a></li>
<li><a href="Triangle_trinomial" title="Triangle trinomial">Triangle trinomial</a></li></ul>
<div class="mw-heading mw-heading3"><h3 id="Liens_externes">Liens externes</h3></div>
<ul><li><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a rel="nofollow" class="external text" href="http://www.4dsolutions.net/ocn/urner.html">Beyond Flatland: Geometry for the 21st Century. PART I: Pascal's Tetrahedron</a></li>
<li><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a rel="nofollow" class="external text" href="http://www.math.rutgers.edu/~erowland/pascalssimplices.html">Pascal's Simplices</a> <i>exposés sur le triangle de Pascal, la pyramide de Pascal, et davantage</i></li></ul>
<ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"></span></span> <span class="bandeau-portail-texte">Portail des mathématiques</span> </span></li> </ul></div><!--htdig_noindex--><div><div class="zim-footer">
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