Pascal's Triangle or Binomial Coefficient
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The
Historic Context
In
1653, a French mathematicianphilosopher 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 Meruprastaara,
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" (杨辉三角形)
by the Chinese. Later, the Persian mathematician Karaji and
the Persian astronomerpoet Omar
Khayyám; thus the triangle is referred to as the "Khayyám
triangle" (مثلث خیام)
in Iran. Several theorems related to the triangle were known,
including the binomial theorem. And to conclude, 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. 
See: Astounding
Pascal's Triangle
