of n letters, and Z the total number of different combinations taken n and n. It is evident, that all the possible arrangements of m letters, taken n at a time, can be obtained, by subjecting the n letters of each of the Z combinations, to all the permutations of which these letters are susceptible. Now a single combination of ʼn letters gives, by hypothesis y permutations; therefore Z combinations will give YxZ... arrangements, taken n and n; and as X denotes the total number of arrangements, it follows that the three quantities X, Y, and Z, give the relations X=YxZ; whence Z= Y' and (Art. 199), Therefore, X=P(m-n+1) P(m-n+1) P m-n+1 == Z= QXn ર n -1 Since P expresses the total number of arrangements, taken nand n-1, and Q the number of permutations of n-1 letters, it follows that P expresses the number of different combinations -1. of m letters taken n-1 and n To apply this to the particular case of combinations of m letters taken 2 and 2, 3 and 3, 4 and 4 . . . ... expresses the number of com binations of the letters taken 2-1 and 2-1 or 1 and 1, and is equal to m; the above formula becomes In like manner, we would find the number of combinations of m m(m-1) (m-2) (m-3) ; and in ge letters taken 4 and 4, to be 1.2.3.4 neral, the number of combinations of m letters taken n and n, is ex Demonstration of the Binomial Theorem. 202. In order to discover more easily the law for the develop ment of the mth power of the binomial x+a, we will observe the law of the product of several binomial factors x+a, x+b, x+c, +d... of which the first term is the same in each, and the second terms different. From these products, obtained by the common rule for algebraic multiplication, we discover the following laws: 1st. With respect to the exponents; the exponent of x, in the first term, is equal to the number of binomial factors employed. In the following terms, this exponent diminishes by unity to the last term, where it is 0. 2d. With respect to the co-efficients of the different powers of x: that of the first term is unity; the co-efficient of the second term is equal to the sum of the second terms of the binomials; the co-efficient of the third term is equal to the sum of the products of the different second terms taken two and two; the co-efficient of the fourth term is equal to the sum of their different products taken three and three. Reasoning from analogy, we may conclude that the co-efficient of the term which has n terms before it, is equal to the sum of the different products of the m second terms of the binomials taken n and n. The last term is equal to the continued pro. duct of the second terms of the binomials. In order to be certain that this law of composition is general, sup. pose that it has been proved to be true for a number m of binomials; let us see if it be true when a new factor is introduced into the product. to be the product of m binomial factors, Nam representing the term which has n terms before it, and Mam-n+1 that which immedi. diately precedes. Let x+K be the new factor, the product when arranged according to the powers of X, will be From which we perceive that the law of the exponents is evident ly the same. With respect to the co-efficients, 1st. That of the first term is unity. 2d. A+K, or the co-efficient of x, is also the sum of the second terms of the m+1 binomials. 3d. B is by hypothesis the sum of the different products of the second terms of the m binomials, and A.K expresses the sum of the products of each of the second terms of the m first binomials, by the new second term K; therefore B+AK is the sum of the different products of the second terms of the m+1 binomials, taken two and two. In general, since N expresses the sum of the products of the se cond terms of the m first binomials, taken n and n; and as MK represents the sum of the products of these second terms, taken n— 1 and n-1, multiplied by the new second term K, it follows that N+MK, or the co-efficient of the term which has n terms before it, is equal to the sum of the different products of the second terms of the m+1 binomials, taken n and n. The last term is equal to the continued product of the m+1 second terms. Therefore, the law of composition, supposed true for a number m of binomial factors, is also true for a number denoted by m+1. It is therefore general. Let us suppose, that in the product resulting from the multiplication of the m binomial factors, x+a, x+b, x+c, x+d... we make a=b=c=d..., the indicated expression of this product, (x+a) (x+b)(x+c), will be changed into (x+a)". With respect to its development, the co-efficients being a+b+c+d..., ab+ac+ad+..., abc+abd+acd. the co-efficient of x-1, or a+b+c+d..., becomes a+a+a+a+ ...9 that is, a taken as many times as there are letters a, b, c and is therefore equal to ma. The co-efficient of 2, or ab+ac+ad+..., reduces to a2+a+a3..., or to a2 taken as many times as we can form different combinations with m-1 m letters, taken two and two, or to m. --a3. (Art. 201). The co-efficient of a reduces to the product of a3, multiplied by the number of different combinations of m letters, taken 3 and In general, if the term, which has n terms before it, is denoted by Na, the co-efficient, which in the hypothesis of the second terms being different, is equal to the sum of their products, taken n and n, reduces, when all of the terms are supposed equal, to a multiplied by the number of different combinations that can be made with m letters, taken n and n. Therefore 203. By inspecting the different terms of this development, a simple law will be perceived, by means of which the co-efficient of any term is formed from the co-efficient of the preceding term. The co-efficient of any term is formed by multiplying the co-efficient of the preceding term by the exponent of x in that term, and dividing the product by the number of terms which precede the required term. For, take the general term P(m-n+1). This is called QXn the general term, because by making n=2, 3, 4 all of the others can be deduced from it. The term which immediately pre ber of combinations of m letters taken n-1 and n-1. Here we P(m-n+1) see that the co-efficient is equal to the co-efficient Qxn |