A Brief History of the Binomial Theorem

Long before the algebraic notation $^{n}C_r$ existed, mathematicians in several ancient civilisations had already worked out the pattern of coefficients that appears when a binomial such as $(a+b)$ is raised to a power.

Ancient India. The earliest known record of the triangular array of these coefficients appears in the Chandah-shastra of Pingala (a treatise on Sanskrit prosody, composed sometime before the 2nd century BCE), where it was used to count the number of ways of arranging long and short syllables in a verse of a given length -- exactly the same counting problem that $^{n}C_r$ solves. The commentator Halayudha (10th century CE) described this array explicitly and named it Meru-prastara ("the expansion of Mount Meru"), giving clear rules for building each row from the row above it.

Ancient China. A similar triangular arrangement of coefficients was described by the mathematician Jia Xian around the 11th century CE, and was later reproduced and popularised in the works of Yang Hui (13th century CE) -- for this reason the array is still commonly called "Yang Hui's triangle" in China today.

The Islamic world. Mathematicians such as al-Karaji (10th-11th century CE) and, following him, Omar Khayyam (11th-12th century CE) studied the expansion of $(a+b)^n$ and its coefficients using combinatorial and inductive arguments, and are credited with early general statements of what we now call the binomial theorem for a positive integer index.

Europe. In the West, the triangular array of coefficients was studied and systematised by Blaise Pascal in his Traite du triangle arithmetique (1654), and it is his name that the array carries in most of the world today, even though -- as the record above shows -- he was far from its first discoverer. The generalisation of the binomial theorem to negative and fractional indices (an infinite series rather than a finite sum) was worked out later still by Isaac Newton around 1665; that extension is beyond the scope of this chapter, which deals only with a positive integral index, exactly as the theorem was known and proved in each of the traditions above.