BINOMIAL THEOREM. 339
407. If in equation (1) of the preceding article we suppose c = d = e = b, we obtain
(a + b)^4 = a^4 + 4a^3b + 6 a^2b^2 + 4 ab^3 + b^4.
We shall now employ the same method to prove a formula known as the Binomial Theorem, by which any binomial of the form a + b can be raised to any assigned positive integral power.
408. To find the expansion of (a + b)^n when n is a positive integer.
Consider the expression
(a + b)(a + c)(a + d)... (a + k)
the number of factors being n.
The expansion of this expression is the continued product of the n factors, a + b, a + c, a + d,... a + k, and every term in the expansion is of n dimensions, being a product formed by multiplying together n letters, one taken from each of these n factors.
The highest power of a is a^n, and is formed by taking the letter a from each of the n factors.
The terms involving a^{n-1} are formed by taking the letter a from any n - 1 of the factors, and one of the letters b, c, d,..., k from the remaining factor; thus the coefficient of a^{n-1} in the final product is the sum of the letters b, c, d,..., k; denote it by S_1.
The terms involving a^{n-2} are formed by taking the letter a from any n-2 of the factors, and two of the letters b, c, d,... k from the two remaining factors; thus the coefficient of a^{n-2} in the final product is the sum of the products of the letters b, c, d,... k taken two at a time; denote it by S_2.
And, generally, the terms involving a^{n-r} are formed by taking the letter a from any n-r of the factors, and r of the letters b, c, d,... k from the r remaining factors; thus the coefficient of a^{n-r} in the final product is the sum of the products of the letters b, c, d,... k taken r at a time; denote it by S_r.
