The manuscript, “Asiatic Research or Transactions of The Society Instituted in Bengal for inquiring into the History and Antiquities, the Arts, Sciences and Literature of Asia – Volume II “, published in the year 1799, has a chapter, “ A Proof that the Hindus had the Binomial Theorem” ( page 487) by Mr Reuben Burrow.
With respect to the Binomial Theorem, the application of it to fractional indices will perhaps remain for ever the exclusive property of Newton, but the following question and its solution evidently show that the Hindoos understood it in whole numbers to the full as well as Briggs, and much better than Pascal. Dr. Hutton, in a valuable edition of Sherwin’s tables, has lately done justice to Briggs; but Mr. Whitchell, who some years before pointed out Briggs as the undoubted inventor of the differential method, said he had found some indications of the Binomial Theorem in much older authors. The method however by which that great man investigated the powers independent of each other is exactly the same as that in the following translation from the Sanscrit.
" A Raja's palace had eight doors; now these doors may either be opened by one at a time, or by two at a time, or by three at a time, and so on through the whole, till at last all are opened together. It is required to tell the numbers of times that this can be done ?
" Set down the number of the doors, and proceed in order, gradually decreasing by one to unity, " and then in a contrary order, as follows :
8 7 6 5 4 3 2 1
1 2 3 4 5 6 7 8
" Divide the first number eight by the unit beneath it, and the quotient eight shows the number of times that the doors can be opened by one at a time. Multiply this last eight by the next term seven, and divide the product by the two beneath it, and the result twenty-eight is the number of times that two different doors may be opened ; multiply the last found twenty-eight by the next figure six, and divide the product by the three beneath it, and the quotient fifty-six shows the number of times that three different doors may be opened. Again, this fifty-six multiplied by the next five, and divided by the four beneath it, is seventy, the number of times that four different doors may be opened. In the same manner fifty-six is the number of fives that can be opened : twenty-eight the number of times that six can be opened: eight the number of times that seven can be opened ; and lastly, one is the number of times the whole may be opened together; and the sum of all the different times is 255."
The demonstration is evident to mathematicians; for as the second term's coefficient in a general equation shows the sum of the roots, therefore, in the n power of 1+1 where every root is unity, the coefficient shows the different ones that can be taken in n things : also, because the third term's coefficient is the sum of the products of all the different twos of the roots, therefore when each root is unity the products of each two roots will be unity, and therefore the number of units, or the coefficient itself, shows the number of different twos that can be taken in n things. Again, because the fourth term is the sum of the products of the different threes that can be taken among the roots, therefore, when each root is unity, the product of each three will be unity, and therefore every unit in the fourth will show a product of three different roots, and consequently the coefficient itself shows all the different threes that can be taken in n things; and so for the rest.