CHAPTER VII.

FORMATION OF POWERS-BINOMIAL THEOREM-EXTRACTION or ROOTS OF ANY DEGREE OF RADICALS.

128. THE solution of equations of the second degree supposes the process for extracting the square root to be known. In like manner, the solution of equations of the third, fourth, &c., degrees, requires that we should know how to extract the third, fourth, &c., roots of any numerical or algebraic quantity.

The power of a number can be obtained by the rules for multiplication, and this power is subject to a certain law of for mation, which it is necessary to know, in order to deduce the root from the power.

Now, the law of formation of the square of a numerical or algebraic quantity, is deduced from the expression for the square of a binomial (Art. 47); so likewise, the law of a power of any degree, is deduced from the expression for the same power of a binomial. We shall therefore first determine the law for the formation of any power of a binomial.

129. By taking the binomial xa several times, as a factor, the following results are obtained, by the rule for multiplication:

(x + a) = x+a,

(x + a)2 = x2 + 2ax + a2,

(;r. + a)3 = x3 + 3ax2 + 3a2x + a3,

(x + α)1 = x2 + 4ax3 + 6a2x2 + 4a3x + a1,

(x + a)5 = x5 + 5ax4 + 10a2x3 + 10a3x2 + 5a1x + a3.

By examining these powers of x+a, we readily discover the luw according to which the exponents of the powers of a de

crease, and those of the powers of a increase, in the successive terms. It is not, however, so easy to discover a law for the formation of the co-efficients. Newton discovered one, by means of which a binomial may be raised to any power, without per forming the multiplications. He did not, however, explain the course of reasoning which led him to the discovery; but the law has since been demonstrated in a rigorous manner. Of all the known demonstrations of it. the most elementary is that which is founded upon the theory of combinations. However, as the demonstration is rather complicated, we will, in order to simplify it, begin by demonstrating some propositions relative to permutations and combinations, on which the demonstration of the binomial theorem depends.

Of Permutations, Arrangements and Combinations.

130. Let it be proposed to determine the whole number of ways in which several letters, a, b, c, d, &c., can be written, one after the other. The result corresponding to each change in the position of any one of these letters, is called a per mutation.

Thus, the two letters a and b furnish the two permutations, ab and ba.

In like manner, the three letters, a, b, c, furnish six permutations.

cab

acb

abc

сось

bca bac

PERMUTATIONS, are the results obtained by writing a certain number of letters one after the other, in every possible order, in such a manner that all the letters shall enter into each result, and each letter enter but once.

To determine the number of permutations of which n letters are susceptible.

Two letters, a and b, evidently give two per

mutations.

ab

I ba

Therefore, the number of permutations of two letters is ex

pressed by 1 x 2.

Reserve

Take the three letters, a, b, and c. either of the letters, as c, and permute the other two, giving

Now, the third letter c may be placed before ab, between a and b, and at the right of ab; and the same for ba: that is, in ONE of the first permutations, the reserved letter c may have three different places, giving three permutations. And, as the same. may be shown for each one of the first permutations, it follows that the whole number of permutations of three letters will be expressed by, 1 × 2 × 3.

ab ba

cab

ach

abc

cba

вси

bac

If, now, a fourth letter d be introduced, it can have four places in each one of the six permutations of three letters : hence, the number of permutations of four letters will be expressed by, 1 × 2 × 3 × 4.

In general, let there be n letters, a, b, c, &c., and suppose the total number of permutations of n-1 letters to be known; and let denote that number,, Now, in each one of the Q permutations, the reserved letter may have n places, giving n permutations hence, when it is so placed in all of them, the entire number of permutations will be expressed by Q × n.

If n = 5, Q will denote the number of permutations of four quantities, or will be equal to 1 × 2 × 3 × 4; hence, the number of permutations of five quantities will be expressed by 1 × 2 × 3 × 4 × 5.

If n = 6, we shall have for the number of permutations of six quantities, 1 x 2 x 3 x 4 x 5 x 6, and so on.

Hence, if y denote the number of permutations of n letters, We shall have

The number of permutations of n letters, is equal to the contrued product of the natural numbers from 1 to n inclusively.

Arrangements.

131. Suppose we have a number m, of letters a, b, c, d, &c. If they are written in sets of 2 and 2, or 3 and 3, or 4 and 4 . . . in every possible order in each set, such results are called

ARRANGEMENTS, are the results obtained by writing a number m

f letters, in sets of 2 and 2, 3 and 3, 4 and 4, n and n; the letters in each set having every possible order, and m being always greater than n.

If we suppose m=n, the arrangements, taken n and n, become permutations.

Having given a number m of letters a, b, c, d, to determine the total number of arrangements that may be formed of them by taking them n in a set.

Let it be proposed, in the first place, to arrange three letters, a, b and C, in sets of two each.

a

b

с

First, arrange the letters in sets of one each, and for each set so formed, there will be two letters reserved: the reserved letters for either arrangement, being those which do not enter it. Thus, with reference to a, the reserved letters are b and c; with reference to b, the reserved letters are a and c; and with reference to c, they are a and b.

Now, to any one of the letters, as a, annex, in succession, the reserved letters b and c : to the second arrangement b, annex the reserved letters a and c; and to the third arrangement, c, annex the reserved letters a and b.

ab

a c

ba

b c

са

c b

Since each of the first arrangements gives as many new arrangements as there are reserved letters, it follows, that the

number of arrangements of three letters taken, two in a set, will be equal to the number of arrangements of the same letters taken one in a set, multiplied by the number of reserved letters.

Let it be required to form the arrangement of four letters, a, b, c and d, taken three in a set.

In sets of two.

ab

a c

ad

ba

bc

b d

ca

First, arrange the four letters in sets of two: there will then be for each arrangement, two reserved letters. Take one of the sets and write after it, in succession, each of the reserved letters: we shall thus form as many sets of three letters each as there are reserved letters; and these sets differ from each other by at least the last letter. Take another of the first arrangements, and annex, in succession, the reserved letters; we shall again form as many different arrangements as there are reserved letters. Do the same for all of the first arrangements, and it is plain, that the whole number of arrangements which will be formed, of four letters, taken 3 and 3, will be equal to the number of arrangements of the same letters, taken two in a set, multiplied by the number of reserved letters.

c b

cd

da

db

dc

In general, suppose the total number of arrangements of m letters, taken n - 1 in a set, to be known, and denote this num

ber by P.

Take any one of these arrangements, and annex to it, in succession, each of the reserved letters, of which the number is

m

(n - 1), or m− n + 1. It is evident, that we shall thus form a number m n+1 of new arrangements of n letters, each differing from the others by the last letter.

1

Now, take another of the first arrangements of n

1 letters, and annex to it, in succession, each of the m — n+1 letters which do not enter it; we again obtain a number m n+ 1 of arrangements of n letters, differing from each other, and from those obtained as above, by at least one of the n - 1 first letters. Now, as we may in the same manner, take all the P arrange ments of the m letters, taken n 1 in a set, and annex to them.