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Pascal's Triangle or Binomial Coefficient

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The Historic Context

In 1653, a French mathematician-philosopher named Blaise Pascal described a triangular arrangement of numbers corresponding to the probabilities involved in flipping coins, or the number of ways to choose 'n' objects from a group of 'm' indistinguishable objects. Pascal's triangle has many uses in binomial expansions. Although Pascal never claimed recognition for his discovery, his name is inseparably linked with it. In fact, the triangle had been described centuries earlier... The first reference occurs in Indian mathematician Pingala's book on Sanskrit poetics that may be as early as 450 BC as Meru-prastaara, the "staircase of Mount Meru". The commentators of this book were also aware that the shallow diagonals of the triangle sum to the Fibonacci numbers. It was also known to Chinese mathematicians. It is said that the triangle was called "Yang Hui's triangle" by the Chinese. Later, the Persian mathematician Karaji and the Persian astronomer-poet Omar Khayyám; thus the triangle is referred to as the "Khayyám triangle" in Iran. Several theorems related to the triangle were known, including the binomial theorem. Finally, in Italy, it is referred to as "Triangolo di Tartaglia" (Tartaglia's triangle), named for the Italian algebraist Niccolò Fontana Tartaglia who lived a century before Pascal; Tartaglia is credited with the general formula for solving cubic polynomials.

The first seven rows of Pascal's Triangle look like:

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1	r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6
Note that every number in the interior of the triangle is the sum of the two numbers directly above it.

Pascal's Arithmetical Triangle The Story of a Mathematical Idea

Related link: Interesting fraction patterns