11. In " Opus novum de proportionibus " he introduced the binomial coefficients and the binomial theorem . 12. Shortly after publishing this paper, de Moivre also generalized Newton's noteworthy binomial theorem into the multinomial theorem. 13. After using the multinomial theorem ( twice the outermost application is the binomial theorem ) and regrouping, 14. This is a very different branch of mathematics from the Binomial Theorem , again showing his impressive intellectual prowess. 15. By virtue of the Binomial Theorem . 16. For larger positive integer values of " n ", the correct result is given by the binomial theorem . 17. :Alternatively, if you expand ( 1-p ) ^ N with the binomial theorem , the first two terms are 1-Np. 18. In between, implicit al-Samaw'al, who used it for special cases of the binomial theorem and properties of Pascal's triangle. 19. Expanding using the binomial theorem and using equation ( 11 ) of the formulas involving binomial coefficients, we obtain 20. The argument supporting the claim that Khayy�m had a general binomial theorem is based on his ability to extract roots.
