Extraction

of roots of higher powers.

(64481201) = 401.

(113028882875) = 4835.

219. The extraction of the roots of higher powers, such as the fourth or fifth, &c. is founded upon

formation of the powers themselves: thus

(a + b)1 = a1 + 4a3b+6a2b2+4ab3 +b2;

the

and if we should propose to pass from this fourth power to its root, the process would stand as follows:

a*+4a3b+6a2b2 + 4ab3 + b* (a+b

a1

4a3) 4a3b+6a2b2 + 4ab3 +b1

4a3b+6a2b2 + 4 ab3 + b1

The first term in the root is the fourth root of the first term in order to determine the second term in the root, divide the first term of the remainder, after subtracting the first term, by four times the cube of the first term of the root: we afterwards, when two terms in the root are determined, form the several terms of the subtrahend, according to the law of formation, of all the terms after the first, of (a+b)*: if there is any remainder, the process may be continued, considering the two terms in the root already determined as one.

It is not necessary, however, to exemplify this process, or to shew in what manner it may be extended to higher roots: for it is very rarely that such operations are required in Algebra and its applications, when the root is finite; and in all other cases, the binomial or multinomial theorem, which we shall investigate in a subsequent chapter, will furnish a more rapid and certain mode of determining them the same remark applies, with greater force, to the extraction of the higher roots of numbers, which are effected by means of logarithms, even in the case of the cube root, more rapidly than by any other methods.

CHAP. IX.

Permuta

tions.

Permutations distinguished from Variations.

Number of

variations

THEORY OF PERMUTATIONS AND COMBINATIONS.

220. THE different orders, in which any quantities can be arranged, are called their Permutations.

Thus the permutations of a and b are ab and ba: the permutations of a, b and c are abc, bac, acb, cab, bea and cba; whilst the permutations of the same three letters, taken two and two together, are ab, ba, ac, ca, bc and cb.

221. The term Permutation is confined by some authors to the different arrangements of the whole of any number of things: whilst the term Variation is applied to the different arrangements of any number of them less than the whole: it may be convenient generally to adopt this distinction, and to use the term Permutation absolutely, inasmuch as the properties of such permutations require more particular attention and examination than those of other classes of variations.

222. We shall now proceed to determine the exof things. pressions for the number of variations of n things, taken two and two, three and three, four and four, and generally r and r together, where r is any number less than n.

Let then things, whose variations are required, be severally represented by a, a, a,...a,, where the numbers subscribed to the same letter a distinguish the things represented from each other, and likewise determine the order of their succession. (Art. 3. and Art. 39. Ex. 22.)

The number of variations, when taken separately or one by one, is clearly n.

and two

223. The number of variations, taken two and two Taken two together, is n (n − 1).

For a, may be placed successively before a, a,,...ɑ„, and thus form (n-1) variations two and two: a, may be placed successively before a1, α ̧, α...α, and thus form (n-1) variations two and two, which are different from the former: and the same thing may be equally done with a, a,...a, there being (n-1) variations corresponding to each letter in the first place, which are different from each other and from all the others (Art. 136): the whole number of such variations is, therefore, n times the number corresponding to each letter in the first place, and is consequently n(n−1).

together.

224. The number of variations, taken three and three Taken three together, is n(n − 1) (n − 2).

For if we form all the variations possible of (n-1) letters (omitting a,), taken two and two together, which are (n-1) (n-2) in number (putting n-1 in the place of n in the expression last determined Art. 223,) and place a, before each of them, we shall have (n-1) (n-2) variations, taken three and three together, where a, occupies the first place and there must be the same number of variations in which a, a,.... and all the other letters successively occupy the first place in each (Art. 136): the whole number, therefore, of such variations, must be n times the number corresponding to each letter in the first place, and is consequently n (n−1) (n-2).

and three

together.

and four

225. By By a similar process of reasoning, we should Taken four shew that the number of variations of n things taken together. four and four together, would be expressed by

formation

where there are four factors, which are the natural num- Law of bers descending from n: and the law which is found to extended by prevail in the formation of the expressions for the number induction to of variations taken two and two, three and three, four ber.

Сс

any num

Its proof.

and four together, may be easily extended by induction to the expression for the number of variations, when any number of things (r) are taken together.

226. In order to demonstrate the correctness of this induction, we must shew, that if this law is true for any one class of variations, it must necessarily be true for the class next superior to it (Art. 139).

Assuming, therefore, the expression for the number of variations of n things taken (r− 1) and (r−1) together, which is

it is required to prove, that the expression for the number of their variations, when they are taken r and r together, is

Omit the first of the letters a,, and form all the variations taken (r−1) and (r−1) together, of the (n-1) letters remaining: the expression for the number of them will clearly be found by putting n-1 in the place of n

* This is the usual mode of denoting the expression for the continued product of a series of terms, whose differences are equal: we write the first, second and last terms, merely interposing a series of dots (in the place of the deficient terms) between the second and the last: the two first terms give us the common difference of the successive terms, and the last determines the extent to which the series of them is carried.

The last term is determined in the following manner: in the first expression (a), the number of terms is (r-1): the first term is n, the second n-1, the third n-2, the fourth n-3, and so on, the number subtracted from n in each term being less by unity, than the number which determines the position of the term in the series: it follows, therefore, that the number which is subtracted from n in the last or (r-1)th term, is r-2: and the last term is, therefore, n―(r-2) or n―r+2. The last term in the product of r such factors, as in the second expression (8), is of course less than n➡r+2 by unity, and is, therefore, n-r+1.

This determination of any assigned term in a series, whose successive terms have equal differences (an arithmetical series), is frequently required in investigations connected with permutations and combinations, as well as other subjects: thus the rth term of such a series, whose first term is n and second n-b, is n—(r−1)b: and if the first term be n and the second n+b, theth term is a +(r−1) b.