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Thus the quantities a, b, c, when taken all together, will form only one com. bination, abc; but will form six different permutations, abc, acb, bac, bca, cab, cba; taken two and two they will form the three combinations ab, ac, bc, and the six permutations ab. ba, ac, ca, bc, cb.

The problem which we propose to resolve is,

190. To find the number of the combinations of n quantities, taken P and p

together.

Let the number of combinations required be x:

Suppose these combinations to be formed and to be written one after the other, in a horizontal line; write below the first of these all the permutations of the p letters which it contains, and since the number of these is 1.2.3......p (=y suppose), we shall have a vertical column consisting of y terms; the second term of the horizontal line will, in like manner, give another vertical column consisting of y terms, being all the permutations of the Р letters which it contains, one at least of which is different from those in the combinations already treated of. The third combination will, in like manner, give y terms differing from all the others. We shall thus form a table consisting of r columns, each of which contains y terms; and on the whole xy results, which are evidently all the permutations of the n letters, taken p and p together, none being either omitted or repeated; we shall therefore have by formula (1),

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Hence we perceive, that the number of the combinations of n quantities, taken p and p together, is equal to the number of the permutations of n quantities, taken P and p together, divided by the number of the permutations of p quantities taken all together.

There is a species of notation employed to denote permutations and combinations, which is sometimes used with advantage from its conciseness.

The number of the permutations of n quantities, taken

......

are represented by
The number of the permutations of n quantities, taken all together,
are represented by

p and P,

(n Pp)

(nPn)

.....(nCp)

The number of the combinations of n quantities, taken p and p,
are represented by

and so on.
It is manifest that the above proposition may be expressed accord
ing to this notation by

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METHOD Of Undetermined COEFFICIENTS.

191. The method of undetermined coefficients is a method for the expan sion or development of algebraic functions into infinite series, arranged according to the ascending powers of one of the quantities considered as a variable. The principle employed in this method may be stated in the following

THEOREM.

If the series A+Bx+Cx2+Dxa3+ &c., whether finite or infinite, be equal to the series A1+B'x+C'x2+D13+ &c., whatever be the value of a; then the coefficients of the like powers of a must be the same in each series; that is, A=A1, B=B', C=C', D=D', &c. ̧

For since

A+B+Ca2+Da3+ &c. =A1+B1x+C1x2+ D1a3+ &c. by transposition we have

(A—A1)+(B—B1)x+(C—C1)x2+(D—D1)x3+ . . . . =0.

Now, if all or any of these coefficients were not =0, the equation would determine particular values of x, and could only be true for such particular values, which is contrary to the hypothesis. Hence we must have A—A1—0, B—B1—0, C—C1=0, &c., and therefore

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This example has been chosen to illustrate the method of expansion by undetermined coefficients; but the development can be obtained by division in the usual way, or by synthetic division, with more facility than by the principle here employed.

(2.) Extract the square root of 1+x.

Assume √1+x=A +Bx+Cx2+DÃ3+ ..., and square both sides;
.. 1+x=A2+ABx+ACx2+ADx3+AEx1+
+ABx+B2x2 + BC3+BDx1+

+AСx2+ BCx3+C2x2 +
+ADa3+BDx+
+A Ea1+

....

....

....

...

hence, equating the coefficients of the like powers of x, we have

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Therefore +=+(!+fx − f2 + 762-821+....)

(3.) Decompose

nominators.

3x-5
x2-13x+40

into two fractions having simple binomial de

By quadratics we find x2-13x+40=(x−5) (x—8); hence we may as

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.. 3x—5—(A+B)x−(8A+5B);

and by the principle of undetermined coefficients we have

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3x-5

=

3

=

64 31 19 1 10 1 x2-13x+40 x-8 x-5 3x-8 3x-5

Note. The values of A and B might have been determined in the following manner:

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Now this equation must subsist for every value of x; and, therefore,

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This method may frequently be employed with advantage, and will be found useful in the integration of rational fractions, when we come to treat of the

Integral Calculus.

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a+c x

Ans. 1−(b+c) = +c (b+c) 2 2 − co (b+c) ~ +...........

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+.....

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PILING OF BALLS AND SHELLS.

(192.) Balls and shells are usually piled in three different forms, called triangular, square, or rectangular, according as the figure on which the pile rests is triangular, square, or rectangular.

(1.) A triangular pile is formed by continued horizontal courses of balls or shells laid one above another, and these courses or rows are usually equilateral triangles whose sides decrease by unity from the bottom to the top row, which is composed simply of one shot; and hence the series of balls composing a triangular pile is

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where n denotes the number of courses in the pile.

(2.) A square pile is formed by continued horizontal courses of shot laid one above another, and these courses are squares whose sides decrease by unity from the bottom to the top row, which is also composed simply of one shot; and hence the series of balls composing a square pile is

1+4+9+16+25+.... n2,

where n denotes the number of courses in the pile.

(3.) The rectangular pile may be conceived to be formed from a square pile, by laying successively on one face of the pyramid a series of triangular strata, cach consisting of as many balls as the face itself contains, and the number of these added triangular strata is always one less than the number of shot in the top row; therefore, if n denote the number of courses, and m+1 the number of shot in the top row, the series composing a rectangular pile is (m+1)+2(m+2)+3(m+3)+4(m+4)+ .... n(m+n)

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(4.) The number of balls in a complete triangular or square pile must evidently depend on the number of courses or rows; and the number of balls in a complete rectangular pile depends on the number of courses, and also on the number of shot in the top row, or the amount of shot in the latter pile depends on the length and breadth of the bottom row; for the number of courses is equal to the number of shot in the breadth of the bottom row of the pile. Therefore, the number of shot in a triangular or square pile is a function of n, and the number of shot in a rectangular pile is a function of n and m.

(5.) If the general term of any series of numbers be of the mth degree, the sum of all the terms of such scries will be of the (m+1)th degree; because

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