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The History of the Binomial Coefficients in India

                                    
        The idea of taking "six tastes one at a time, two at a time, three at a time, etc." was written down correctly in India 300 years before the birth of Christ in a book called the Bhagabati Sutra, a text from the Jainist religion; this gives the subcontinent of India the distinction of being the earliest civilization to have an understanding of the binomial coefficients in their combinatorial form "n choose k" in a text that survives to this day.

        Prosody, the study of rhythm and meter in songs and poetry, was the interest of Pingala (circa 200 B.C.), who called his rule Meru Prastara; instead of tastes, he was now thinking about six syllables in a poem, which could be any combination of long and short.  While his original text does not fully survive, surviving commentaries on his work show that he understood the additive rule.  Among the surviving commentaries on the Meru Prastara are the work of Varahamihara (505 A.D.), which talks about the additive rule, and Bhattotpala (1068 A.D.) has a table which correctly lists the number of ways to take two things at a time, three things at a time, four things at a a time, etc. from sets as big as sixteen things.

        Another Jain mathematician Mahavira (circa 850 A.D.) writing in Ganita Sara Sangraha completely generalized the rule found in the Bhagabati Sutra, written over a millenium previously.

        The great Hindu mathematician Bhaskara (circa 1100 A.D.) repeats Mahavira's work in his Lilavati, which is more accessible to Western readers, and includes the idea of multiplicative expansion of a row of the triangle, an idea Edwards could not find in the Chinese texts which put the numbers in their well-known triangular form.

        Bhaskara's particular interest, like Pingala's, was prosody.  In this context, Bhaskara uses the following sequence of fractions: .  The idea is this:  there is only one way to have six short syllables; from there, we multiply by 6 to get the number of ways to do five short and one long, then multiply that result by ⁵/₂ to get the number of ways to to do four short and two long, and continuing down the line we get the sixth row 1, 6, 15, 20, 20, 15, 6, 1.  Bhaskara also understood the idea of the multinomial coefficient in reference to arranging digits and/or letters; this was work original to him that did not appear in Mahavira.

        The idea of expansion of a binomial was not well studied in India, though Brahmagupta (628 A.D.) had correctly expanded (a+b)³, one level higher than is found in the surviving work of the great Greek mathematician Euclid.  While Brahmagupta's work is not the greatest achievement of Indian mathematics, there is evidence that it made its way to Baghdad two centuries later, and may well be the seed from which the tree of knowledge in the Middle East about the binomial coefficients first grew.

Jainist religion The Jains believe in the liberation of the soul by right faith, right knowledge and right conduct.  While their numbers are not as large as the Hindus or Buddhists, the religion survives to this day and has adherents around the world.

Prosody  The problem studied by Pingala is similar to the long and short clicks used in Morse code; in music from India, the number of actual beats is counted, instead of thinking about the number of beats of a certain length, as is used in Western musical notation such as 4/4 or 6/8.  For example, Indian musicians would categorize the first bar of the 4/4 rock song Louie, Louie as a six beat pattern, short-short-short-long-short-long, where the beat in italics indicate a rest.  This is continued ad nauseum or until the band takes a break, whichever comes first.

(The author would like to apologize to the memory of the late Richard Berry, the songwriter of Louie, Louie, for that last joke.  The author met Mr. Berry over 20 years ago at an all-weekend Louie, Louie marathon at radio station KFJC in Los Altos Hills, and Mr. Berry was very gracious indeed.)