Name: pascal’s triangle for binomial coefficients
Uploaded: 2025-11-26T05:43:48.000Z
Duration: 1 min 30 s
Channel: JoVE

Un binôme est une expression de la forme a + b, où a et b sont des nombres ou des expressions algébriques.L’élever à une puissance n produit une série de termes qui suivent un modèle prévisible.Chaque développement a n + 1 termes, commençant par a n et se terminant par bn.Ces motifs s’alignent sur un outil visuel connu sous le nom de Triangle de Pascal.Le triangle de Pascal est un tableau triangulaire où chaque ligne fournit les coefficients d’un binôme élevé à une puissance spécifique.Par exemple, la cinquième ligne donne les coefficients de a + b élevé à la puissance cinq.Chaque rangée commence et se termine par un, et chaque nombre intérieur est égal à la somme des deux nombres en diagonale au-dessus. Le triangle de Pascal fournit les coefficients du théorème binomial, calculés comme suit n choisissez k.Ce modèle s’applique également aux probabilités. Dans les lancers de pièces, H et T représentent pile et face. Pour trois lancers, la somme de H et T élevée à la puissance trois représente tous les résultats possibles.Après avoir développé et comparé avec le triangle de Pascal, chaque terme correspond à un résultat possible : trois faces, deux têtes et une queue, une tête et deux queues, ou trois queues.

développement binomial à l’aide du triangle de pascal

Binomial Expansion Using Pascal's Triangle

Le d&#233;veloppement d&#8217;une expression binomiale telle que (a + b)^n produit une suite pr&#233;visible de termes, que l&#8217;on peut obtenir syst&#233;matiquement &#224; l&#8217;aide du triangle de Pascal. Ce tableau triangulaire de nombres joue un r&#244;le essentiel dans la compr&#233;hension et le calcul des coefficients des d&#233;veloppements binomiaux.Le triangle de Pascal est construit de telle sorte que chaque ligne corresponde aux coefficients d&#8217;un bin&#244;me &#233;lev&#233; &#224; une puissance. La ligne sup&#233;rieure, dite &#171; rang 0 &#187;, correspond &#224; (a + b)^0, et chaque ligne successive fournit les coefficients pour des puissances croissantes de n. Par exemple, la ligne de rang 5 du triangle de Pascal, 1, 5, 10, 10, 5, 1, donne les coefficients du d&#233;veloppement de (a + b)^5. Chaque ligne commence et se termine par le nombre 1. Chaque coefficient int&#233;rieur est obtenu en additionnant les deux coefficients situ&#233;s en diagonale au-dessus de lui dans la ligne pr&#233;c&#233;dente, ce qui illustre la structure r&#233;cursive du triangle.Dans le d&#233;veloppement de (a + b)^n, les exposants de a d&#233;croissent de n &#224; 0, tandis que ceux de b augmentent de 0 &#224; n. Par cons&#233;quent, chaque terme du d&#233;veloppement prend la forme suivante :<figure class='image image-style-align-center image_resized' style='display:table;margin-left:auto;margin-right:auto;width:75%;'><img style='aspect-ratio:300/22;' src='https://cloudfront.jove.com/files/ftp_upload/41037/41039_Equation_editor_41037.svg' alt='41039_Equation.svg' /></figure>O&#249; &#171; n parmi r &#187; d&#233;signe le coefficient binomial, trouv&#233; &#224; la r-i&#232;me position de la n-i&#232;me ligne dans le triangle de Pascal.

Développement binomial à l’aide du triangle de Pascal

Le d&#233;veloppement d&#8217;une expression binomiale telle que (a + b)^n produit une suite pr&#233;visible de termes, que l&#8217;on peut obtenir ...

binomial-expansion-using-pascal-s-triangle

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pascal’s triangle for binomial coefficients

Pascal’s Triangle helps organize the coefficients in a binomial expansion. A binomial is an expression with two terms, such as (a + b), and raising it to a power creates a predictable pattern of terms.

The triangle is built row by row. The top row, called the zeroth row, matches (a + b)0. Each new row gives the coefficients for the next power of the binomial, so the sixth row, 1, 5, 10, 10, 5, 1, gives the coefficients for (a + b)5.

Each row begins and ends with 1. Every number inside the triangle is found by adding the two numbers directly above it, one from each diagonal. This recursive pattern makes the triangle easy to extend.

In the expansion of (a + b)n, the power of a decreases from n to 0. At the same time, the power of b increases from 0 to n. Each term uses a binomial coefficient, written as “n choose r,” and that coefficient is found at the rth position in the nth row of Pascal’s Triangle.

Pascal’s Triangle for Binomial Coefficients

pascals-triangle-for-binomial-coefficients

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binomial expansion using pascal's triangle

Pascal’s Triangle helps organize the coefficients in a binomial expansion. A binomial is an expression with two terms, such as (a + b), and raising it to a power creates a predictable pattern of terms. The triangle is built row by row. The top row, called the zeroth row, matches (a + b)0. Each new row gives the coefficients for the next power of the binomial, so the sixth row, 1, 5, 10, 10, 5, 1, gives the coefficients for (a + b)5. Each row begins and ends with 1. Every number inside the...

key terms and concepts

Learn how Pascal’s Triangle gives the coefficients for binomial expansions, with rows, pattern rules, and the meaning of “n choose r.”

Pascal’s Triangle for Binomial Coefficients | JoVE K12

The JoVE Math Fundamentals collection is designed to build a deep and comprehensive understanding of essential mathematics. Covering the foundations of algebra, geometry, trigonometry, and analytic reasoning, this course establishes a strong base for future studies. The collection connects core mathematical concepts with real-world practice, equipping learners to solve complex problems, master problem-solving skills, and apply abstract ideas.

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Pascal’s Triangle for Binomial Coefficients

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