right|thumb|250px|the first six rows of pascal's triangle in mathematics, pascal's triangle is a triangular array of the binomial coefficients. in much of the western world it is named after french mathematician blaise pascal, although other mathematicians studied it centuries before him in india,maurice winternitz, history of indian literature, vol. iii iran, china, germany, and italy. the rows of pascal's triangle are conventionally enumerated starting with row n&nbsp;=&nbsp;0 at the top. the entries in each row are numbered from the left beginning with k&nbsp;=&nbsp;0 and are usually staggered relative to the numbers in the adjacent rows. a simple construction of the triangle proceeds in the following manner. on row 0, write only the number&nbsp;1. then, to construct the elements of following rows, add the number above and to the left with the number above and to the right to find the new value. if either the number to the right or left is not present, substitute a zero in its place. for example, the first number in the first row is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in the third row are added to produce the number 4 in the fourth row. this construction is related to the binomial coefficients by pascal's rule, which says that if :(x+y)^n=\sum_{k=0}^n{n \choose k}x^{n-k}y^{k} then : {n \choose k} = {n-1 \choose k-1} + {n-1 \choose k} for any non-negative integer n and any integer k between 0 and n.the binomial coefficient \scriptstyle {n \choose k} is conventionally set to zero if k is either less than zero or greater than n. pascal's triangle has higher dimensional generalizations. the three-dimensional version is called pascal's pyramid or pascal's tetrahedron, while the general versions are called pascal's simplices. thumb|each number in the triangle is the sum of the two directly above it. ==history==  thumb|right|blaise pascal's version of the triangle the set of numbers that form pascal's triangle were known before pascal. however, pascal developed many uses of it and was the first one to organize all the information together in his treatise, traitÃ© du triangle arithmÃ©tique (1653). the numbers originally arose from indian studies of combinatorics and binomial numbers and the greeks' study of figurate numbers.[http://www.bookrags.com/research/pascals-triangle-wom/ pascal's tria
Blaise_Pascal Chinese_mathematical_discoveries Factorial_and_binomial_topics Indian_inventions Triangles_of_numbers