® The coefficients of the expansions are arranged in an array. This array is

® The coefficients of the expansions are arranged in an array. This array is called Pascal’s triangle . ® The general term of an expansion ( a + b ) n is T r + = n C r a n – r . b r .

® In the expansion ( a + b ) n , if n is even, then the middle term is the th term.If n is odd, then the middle terms are th n + and th n + terms. Historical Note The ancient Indian mathematicians knew about the coefficients in the expansions of ( x + y ) n , ≤ n ≤ . The arrangement of these coefficients was in the form of a diagram called Meru-Prastara , provided by Pingla in his book Chhanda shastra (200B.C.). This triangular arrangement is also found in the work of Chinese mathematician Chu-shi-kie in .

The term binomial coefficients was first introduced by the German mathematician, Michael Stipel ( - ) in approximately . Bombelli ( ) also gave the coefficients in the expansion of ( a + b ) n , for n = , ..., and Oughtred ( ) gave them for n = , ,..., . The arithmetic triangle, popularly known as Pascal’s triangle and similar to the Meru- Prastara of Pingla was constructed by the French mathematician Blaise Pascal ( - ) in . The present form of the binomial theorem for integral values of n appeared in Trate du triange arithmetic , written by Pascal and published posthumously in .