∑ A diagram that shows Pascal's triangle with rows 0 through 7. 5 3 A similar pattern is observed relating to squares, as opposed to triangles. In mathematics, Pascal's triangle is a triangular array of the binomial coefficients. × n {\displaystyle (a+b)^{n}=b^{n}\left({\frac {a}{b}}+1\right)^{n}} The triangle was later named after Pascal by Pierre Raymond de Montmort (1708) who called it "Table de M. Pascal pour les combinaisons" (French: Table of Mr. Pascal for combinations) and Abraham de Moivre (1730) who called it "Triangulum Arithmeticum PASCALIANUM" (Latin: Pascal's Arithmetic Triangle), which became the modern Western name. {\displaystyle 0\leq k\leq n} x An alternative formula that does not involve recursion is as follows: The geometric meaning of a function Pd is: Pd(1) = 1 for all d. Construct a d-dimensional triangle (a 3-dimensional triangle is a tetrahedron) by placing additional dots below an initial dot, corresponding to Pd(1) = 1. + {\displaystyle (x+1)^{n}} In the 13th century, Yang Hui (1238–1298) presented the triangle and hence it is still called Yang Hui's triangle (杨辉三角; 楊輝三角) in China. 1 Since the columns start with the 0th column, his x is one less than the number in the row, for example, the 3rd number is in column #2. + n -terms are the coefficients of the polynomial n 0 ( In this case, we know that Any pictures i could find on the web go up to the 20th row, but when printed you cannot see the numbers. 1 0 + There are some patterns to be noted.1. {\displaystyle n} 7 {\displaystyle n} n ( ( , ..., and the elements are = ) ( [7] Gerolamo Cardano, also, published the triangle as well as the additive and multiplicative rules for constructing it in 1570. Let's begin by considering the 3rd line of Pascal's triangle, with values 1, 3, 3, 1. equal to one. (a) Find the sum of the elements in the first few rows of Pascal's triangle. 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