Elements of Algebra

CHAP. XI. OF ALGEBRA. 119 also how to find all the powers of roots composed of more than two terms *. We shall apply them to the third power of $a + b + c$; the terms of which must be formed by all the possible combinations of three letters, each term having for its coefficient the number of its transpositions, as shewn, Art. 352. Here, without performing the multiplication, the third power of $(a + b + c)$ will be, $a^3 + 3a^2b + 3a^2c + 3ab^2 + 6abc + 3ac^2 + b^3 + 3b^2c + 3bc^2 + c^3$. Suppose $a = 1$, $b = 1$, $c = 1$, the cube of $1 + 1 + 1$, or of $3$, will be $1+3+3+3+6+3+1+3+3+1=27$; Let us take the formula $abc$; here we know that the letters which compose it admit of six permutations, namely, $abc, acb, bac, bca, cab, cba$: but as for combinations, it is evident that by taking these three letters one by one, we have three combinations, namely, $a, b$, and $c$; if two by two, we have three combinations, $ab, ac$, and $bc$; lastly, if we take them three by three, we have only the single combination $abc$. Now, in the same manner as we prove that $n$ different things admit of $1 \times 2 \times 3 \times 4 \cdots n$ different permutations, and that if $r$ of these $n$ things are equal, the number of permutations is $\frac{1 \times 2 \times 3 \times 4 \cdots n}{1 \times 2 \times 3 \times \cdots r}$; so likewise we prove that $n$ things may be taken $r$ by $r$, $\frac{n \times (n-1) \times (n-2) \cdots (n-r+1)}{1 \times 2 \times 3 \cdots r}$ number of times; or that we may take $r$ of these $n$ things in so many different ways. Hence, if we call $n$ the exponent of the power to which we wish to raise the binomial $a + b$, and $r$ the exponent of the letter $b$ in any term, the coefficient of that term is always expressed by the formula $\frac{n \times (n-1) \times (n-2) \cdots (n-r+1)}{1 \times 2 \times 3 \cdots r}$. Thus, in the example, article 359, where $n = 7$, we have $a^5b^2$ for the third term, the exponent $r = 2$, and consequently the coefficient $=\frac{7 \times 6}{1 \times 2}$; for the fourth term we have $r = 3$, and the coefficient $=\frac{7 \times 6 \times 5}{1 \times 2 \times 3}$; and so on; which are evidently the same results as the permutations. For complete and extensive treatises on the theory of combinations, we are indebted to Frenicle, De Montmort, James Bernoulli, &c. The two last have investigated this theory, with a view to its great utility in the calculation of probabilities. F. T. * Roots, or quantities, composed of more than two terms, are called *polynomials*, in order to distinguish them from *binomials*, or quantities composed of two terms. F T.