COR. Hence also the number of terms in the expansion of any multia,)", is known; for it is obviously the same as the number of homogeneous products of r things taken n together, that is, nomial, as (a, + A2 + A3 + If r = 2, that is for a binomial, (a+b)", the expression becomes 2.3.4 (n + 1) 1.2.3 ...... or n + 1. If r = 4 or the quantity to be expanded be (a+b+c+d)", the number of terms is 4.5.6 (n + 3) or and so on for any value of r. THE EXPONENTIAL THEOREM. 325. To expand a in a series of powers of x. a" = { (1 + a − 1)"}"; and expanding by the Binomial Theorem, = {1 + An + Bn® + Cn2 + &c.}", if a − 1 − ¦ (a − 1)2 + &c. = 4 A Now, since a is clearly independent of n, n may be any value whatever; let, then, n = 0; COR. If e be that value of a which makes A equal to 1, then 326. The Multinomial Theorem is a rule or formula for expanding any power of a quantity which consists of more than two terms. The expansion of a multinomial may frequently be effected by the Binomial Theorem, as is done for a trinomial in Art. 322; for (a+b+c+d+&c.) may be expanded as a binomial by considering any number of terms as one term, and the remainder as another term. But a more general method is to find the general term, and to deduce the whole expansion from that term as follows: 327. To find the general term of the expansion of (a+b+c+d+&c.)m. Let b+c+d+ &c.=z, then (a + b + c + d + &c.)′′ = (a + z)", of which the general term, expressed by the a+1th, is a (a − 1 ) ...... ( a − 9 + 1 ) b1y3, (where q + ß = a, or p + q + ß=m), 1 2 ...... q ; so that the general term of the multinomial becomes byB Again, if d+ &c. = x, y = (x + c), of which the general term, expressed by the r+1th, is ; so that the general term of the multinomial becomes and so on, until the terms of the multinomial are exhausted. Hence, the general term required is p being fractional or negative when m is fractional or negative, but q, r, s, &c. always positive integers. COR. If m is a positive integer, then, since p is a positive integer, the expression for the general term may be written The last result may also be arrived at by the following method, assuming the index a positive integer: : 328. To expand (a+b+c+d+&c.)TM, when m is a positive integer. e €(a+b+c+d+&c.)x = Єax. Єbx. Єcx. edx. &c. and if € = 2.7182818, expanding by the Exponential Theorem, Ꮖ 1 + (a + b + c + d + &c.) + ( a + b + c + d + &c.)3 12 + ... |2 Now, as this operation merely exhibits the same quantity expanded in two different ways by the same theorem, the corresponding terms, that is, the terms involving the same powers of x will be equal to each other; therefore equating the coefficient of a" on the one side with the coefficient of a" on the other, and observing that each separate term on this side of the equation which involves am will be the product of as many terms as there are series to be multiplied, one of which is taken out of each series, and will therefore be of the form COR. 1. If q+r+s+&c.= π, then p=m-π, and the general term becomes which form is sometimes found more convenient. COR. 2. If it be required to expand (a,+a, x + a‚x2 + A3x3 + &c.)", the general term may be obtained from that of (a + b + c + d + &c.)" by writing a, a,x, a,x2, ax3, &c. in place of a, b, c, d, &c. respectively, by which it becomes and all the terms of the expansion may be found as before, by giving p, q, r, s, &c. all possible values which the condition p + q +r+s+ &c. admits of. = m Also any particular term involving a proposed power of x, as a", will be found by taking the sum of the values of this general term, when p, q, r, s, &c. are made to assume all the values, which satisfy the two equations p+q+r+s+&c.=m, and q+2r+3s + &c. = n. COR. 3. Assuming the Theorem for a positive integral index, it may be proved for a fractional or negative index thus: Σ stands for the expression "the sum of all the quantities of the form of." + The proof here given of the Multinomial Theorem extends only to the case of positive integral indices, for by the Exponential Theorem m cannot be any thing but a positive integer. But if the Multinomial be deduced from the Binomial Theorem, (as in Art. 327) then since the latter is proved for fractional and negative indices, the former is also proved to hold for such indices. Let b+c+d+ &c. = x, then (a+b+c+d+ &c.)TM = (a1+ x)", of which the general term, expressed by the a+1th, is and since a is a positive integer, by what has been proved the general Ex. 1. Required the term in the expansion of (a-b-c) which involves a b3 c2. Ex. 2. Required the term in the expansion of (a + bx + cx2 + dx3)1 which involves x8. +2r+ 3s q = where p+q+r+s= 4, and 8; and it remains to find all the values of p, q, r, s which satisfy these equations. To do this, it is most convenient to take in order, beginning with the highest or the lowest, the several values of q, r, and s, which satisfy the latter, and reject those which are inconsistent with the former, equation. Thus beginning with q, we see that q cannot be greater than 4, therefore we try this value, which does not answer: then q=3, which also does not answer: then 9=2, which answers with p=0, r = 0, s=2; then q = 1, which answers with p=0, r = 2, s = 1; then q=0, which answers with p= 1, r = 1, s=2. Next try r = 4, which answers with p=0, q=0, s=0: then r = 3, which does not answer; then r = 2, r = 1, r =0, which will not answer except as already determined. Next try s = 4, 3, 2, 1, 0, in order; and no other values will answer except those already mentioned. The result, then, stands thus |