Without carrying these products any further, we may discover the law according to which they are formed.

By supposing all the terms involving the same power of x, and placed in the same column, to form only one, as, for example,

a x3 + bx3 + c x3 + d x3 = (a + b + c + d) x3,

&c.

(1.) We find in each product one term more than there are units in the number of factors.

(2.) The exponent of x in the first term is the same as the number of factors, and goes on decreasing by unity in each of the following terms.

(3.) The greatest power of x has unity for its coefficient; the following, or that, whose exponent is one less, is multiplied by the sum of the second terms of the binomials; that, whose exponent is two less, is multiplied by the sum of the different products of the second terms of the binomials taken two and two; that, whose exponent is three less, is multiplied by the sum of the different products of the second term of the binomials, taken three and three, and so on; in the last term, the exponent of x, being considered as zero (37), is equal to that of the first diminished by as many units as there are factors employed, and this term contains the product of all the second terms of the binomials.

It is manifest, that the form of these products must be subject to the same laws, whatever be the number of factors; as may be shown by other evidence beside that from analogy.

136. It will be seen immediately, that the products, of which we are speaking, must contain the successive powers of x, from that, whose exponent is equal to thenumber of factors employed, to that, whose exponent is zero. To present this proposition under a general form, we shall express the number of factors by the letter m; the successive powers of x will then be denoted by

xm, xm-1, xm-2, &c.

Y,

We shall employ the letters, A, B, C, to express the quantities, by which these powers, beginning with -1, are to be multiplied; but as the number of terms, which depends on the particular value given to the exponent, will remain indeterminate, so long as this exponent has no particular value, we can write only the first and last terms of the expression, designating the intermediate terms by a series of points.

the formula then

.....

xm + A xm−1+В xm-2 + C xm-3 represents the product of any number m of factors, x + a, x + b, x + c, x + d, &c.

+Y,

If we multiply this by a new factor x + 1, it becomes

1

xm 1+ Axm + Bxm-1 + C xm-2

+1xm +1 Axm¬1+1 B x3

-2

It is evident, 1. that if A is the sum of the m second terms a, b, c, d, &c. A+ will be that of the m + 1 second terms a, b, c, d, &c. l, and that consequently the expression employed to denote the coefficient will be true for the product of the degree m+1, if it is true for that of the degree m.

2. If B is the sum of the products of the m quantities a, b, c, d, &c. taken two and two, B+1A will express that of the products of the m+1 quantities a, b, c, d, &c. l, taken also two and two; for A being the sum of the first, IA will be that of their products by the new quantity introduced 7; therefore the expression employed will be true for the degree m+1, if it is for the degree m.

If C is the sum of the products of the m quantities a, b, c, d, &c. taken three and three, C+1B will be that of the products of the m+1 quantities a, b, c, d, &c. l, taken also three and three, since 1B, from what has been said, will express the sum of the products of the first taken two and two, multiplied by the new quantity introduced l; therefore, the expression employed will be true for the degree m+1, if it is true for the degree m.

It will be seen, that this mode of reasoning may be extended to all the terms, and that the last, Y will be the product of m +1 second terms.

The propositions laid down in art. 135., being true for expressions of the fourth degree, for example, will be so, according to what has just been proved, for those of the fifth, for those of the sixth, and, being extended thus from one degree to another, they may be shown to be true generally.

It follows from this, that the product of any number whatever m, of binomial factors x+a, x + b, x + c, x + d, &c. being represented by

xm + A xm¬1 + В xm−2+ C xm-3 + &c.

A will always be the sum of the m letters a, b, c, &c., B that of the products of these quantities, taken two and two, C that of the products of the quantities, taken three and three, and so on.

To comprehend the law of this expression in a single term, I take one, whose place is indeterminate, and which may be represented by Nam-n.

This term will be the second, if we make n = 1, the third, if we' make n = 2, the eleventh, if we make n = 10, &c. In the first case, the letter N will be the sum of the m letters, a, b, c, &c., in the second, that of their products, when taken two and two; in the third, that of their products, when taken ten and ten; and in general, that of their products, taken n and n.

All the quantities, by which the same power of x is multiplied, become in this case equal; thus the coefficient of the second term, which in the product

(x + a) (x + b) (x + c) (x + d) is a + b + c + d, is changed into 4a; that of the third term in the same product, which is,

ab + ac+ad + b c + b d + cd,

becomes 6 a2. Hence it is easy to see, that the coefficients of the different powers of x will be changed into a single power of a, repeated as many times as there are terms, and distinguished by the number of factors contained in each of these terms. Thus, the coefficient N, by which the power am-n is multiplied, will, in the general developement, be that power of a denoted by n, or a", repeated as many times, as we can form different products by taking in every possible way a number n of letters from among a number m; to find the coefficient of the term containing - then is reduced to finding the number of these products.

138. In order to perform the problem just mentioned, it is necessary to distinguish arrangements or permutations from products or combinations. Two letters, a and b, give only one product, but admit of two arrangements, ab and ba; three letters, a, b, c, which give only one product, admit of six arrangements (88), and so on.

To take a particular case, I will suppose the whole number of letters to be nine, namely,

a, b, c, d, e, f, g, h, i,

and that it is required to arrange them in sets of seven. It is evident, that if we take any arrangement we please, of six of these letters, a b c d e f, for example, we may join successively to it each of the three remaining letters, g, h, and i; we shall then have three arrangements of seven letters, namely,

abcdefg, abcdefh,

abcdefi.

What has been said of a particular arrangement of six letters, is equally true of all; we conclude, therefore, that each arrangement of six letters will give three of seven, that is, as many as there remain letters, which are not employed. If, therefore, the number of arrangements of six letters be represented by P, we shall obtain the number consisting of seven letters by multiplying P by 3 or 96. Representing the numbers 9 and 7 by m and n, and regarding P as expressing the number of arrangements, which can be furnished by m letters, taken n I at a time, the same reasoning may be employed; we shall thus have for the number of arrangements of n letters,

P (m − (n − 1)),

or P (mn+1).

This formula comprehends all the particular cases, that can occur in any question. To find, for example, the number of arrangements, that can be formed out of m letters, taken two and two, or two at a time, we make n = 2, which gives

for P will in this case be equal to the number of letters taken one at a time; there results then from this

1).

m (m2 + 1), or m (m for the number of arrangements taken two and two. Again, taking

P = m (m − 1) and n = 3,

we find for the number of arrangements, which m letters admit of, taken three and three,

m n (m − 1) (m − 3 + 1) = m (m − 1) (m — 2).

Making

P = m (m − 1) (m — 2) and

n = 4,

for the number of arrangements taken four and four. We may thus determine the number of arrangements, which may be formed from any number whatever of letters.*

139. Passing now from the number of arrangements of n letters, to that of their different products, we must find the number of arrangements, which the same product admits of. In order to this, it may be observed, that if in any of these arrangements, we put one of the letters in the first place, we may form of all the others as many permutations, as the product of n — 1 letters admits of. Let us take, for example, the product abcdefg, composed of seven letters; we may, by putting a in the first place, write this product in as many ways, as there are arrangements in the product of six letters bcdefg; but each letter of the proposed product may be placed first. Designating then the number of arrangements, of which a product of six letters is susceptible, by Q, we shall have 7 for that of the arrangements of a product of seven letters. It follows from this, that if Q designate the number of arrangements, which may be formed from a product of n1 letters, Qn will express the number of arrangements of a product of n letters.

* In these arrangements it is supposed by the nature of the inquiry that there are no repetitions of the same letter; but the theory of permutations and combinations, which is the foundation of the doctrine of chances, embraces questions in which they occur. The effect may be seen in the example we have selected, by observing, that we may write indifferently each of the 9 letters a, b, c, d, e, f, g, h, i, after the product of 6 letters abcdef. Designating, therefore, the number of arrangements, taken six at a time, by P, we shall have P × 9 for the number of arrangements, taken 7 at a time. For the same reason, if P denote the number of arrangements of m letters, taken n 1 at a time, that of their arrangements, when taken n at a time, will be P m.

This being admitted, as the number of arrangements of m letters, taken one at a time, is evidently m, the number of arrangements, when taken 2 and 2, will be m × m, or m2, when taken 3 and 3, the number will be m× mm, or m3; and lastly, m" will express the number of arrangements, when they are taken n and n.