In mathematics, Pascal's triangle is a triangular array of the binomial coefficients. It is named after the French mathematician Blaise Pascal in much of the Western world, although other mathematicians studied it centuries before him in India, Greece, Iran, China, Germany, and Italy.
The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top. The entries in each row are numbered from the left beginning with k = 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 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 0 + 1 = 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
then
for any nonnegative integer n and any integer k between 0 and 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.
Read more about Pascal's Triangle: History, Binomial Expansions, Combinations, Relation To Binomial Distribution and Convolutions, Extensions
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“There are only three kinds of people: those who serve God, having found him; others who are occupied in seeking him, not having found him; while the remainder live without seeking him and without having found him. The first are reasonable and happy; the last are foolish and unhappy; those between are unhappy and unreasonable.”
—Blaise Pascal (16231662)