Elements of Algebra

CHAP. XI. OF ALGEBRA. 115 1st coefficient = 1; 2d = $1 \cdot \frac{10}{1} = 10$; 7th = $252 \cdot \frac{5}{6} = 210$; 3d = $10 \cdot \frac{9}{2} = 45$; 8th = $210 \cdot \frac{4}{7} = 120$; 4th = $45 \cdot \frac{8}{3} = 120$; 9th = $120 \cdot \frac{3}{8} = 45$; 5th = $120 \cdot \frac{7}{4} = 210$; 10th = $45 \cdot \frac{2}{9} = 10$; 6th = $210 \cdot \frac{6}{5} = 252$; and 11th = $10 \cdot \frac{1}{10} = 1$. 351. We may also write these fractions as they are, without computing their value; and in this manner it is easy to express any power of $a + b$. Thus, $(a + b)^{100} = a^{100} + \frac{100}{1} a^{99}b + \frac{100 \cdot 99}{1 \cdot 2} a^{98}b^2 + \frac{100 \cdot 99 \cdot 98}{1 \cdot 2 \cdot 3} a^{97}b^3$ $+ \frac{100 \cdot 99 \cdot 98 \cdot 97}{1 \cdot 2 \cdot 3 \cdot 4} a^{96}b^4 +$, \ldots * Whence the law of the succeeding terms may be easily deduced. CHAP. XI. *Of the Transposition of the Letters, on which the demonstration of the preceding Rule is founded.* 352. If we trace back the origin of the coefficients which we have been considering, we shall find, that each term is presented as many times as it is possible to transpose the letters of which that term is composed; or, to express the same thing differently, the coefficient of each term is equal to the number of transpositions which the letters composing that term admit of. In the second power, for example, the term $ab$ is taken twice, that is to say, its coefficient is 2; and in fact we may change the order of the letters which compose that term twice, since we may write $ab$ and $ba$. * Or, which is a more general mode of expression, $(a + b)^n = a^n + \frac{n}{1}a^{n-1}b + \frac{n \cdot (n-1)}{1 \cdot 2}a^{n-2}b^2$ $+ \frac{n \cdot (n-1) \cdot (n-2)}{1 \cdot 2 \cdot 3}a^{n-3}b^3 + \frac{n \cdot (n-1) \cdot (n-2) \cdot (n-3)}{1 \cdot 2 \cdot 3 \cdot 4}$ $a^{n-4}b^4$ \ldots $--- \frac{n \cdot (n-1) \cdot (n-2) \cdot (n-3) \ldots 1}{1 \cdot 2 \cdot 3 \cdot 4 \ldots n}$ This elegant theorem for the involution of a compound quantity of two terms, evidently includes all powers whatever; and we shall afterwards shew how the same may be applied to the extraction of roots. I 2