Writing a before each of these (n-1) (n−2) (n−3) ... (n—p+1) permutations, we shall have (n-1)(n—2)(n—3)......................... (n-p+1) permutations of the n quantities, in which a stands first. Reasoning in the same manner for b, we shall have (n−1)(n—2)(n—3)... ... ... ... ... ... ... ... ... ... . . (n−p+1) permutations of the n quantities, in which b stands first; and so on for each of the n quantities in succession; hence the whole number of permutations will be (n−p+1) . . . . . (1) n(n−1)(n—2)(n—3). . . Hence it appears that, if the above law of formation hold good for any one class of permutations, it must hold good for the class next superior; but it has been proved to hold good when p=2, or for the permutations of n quantities taken two and two; hence it must hold good when p=3, or for the permutation of n quantities taken three and three; .. it must hold good when p=4, and so on. The law is, therefore, general. EXAMPLE. Required the number of the permutations of the eight letters a, b, c, d, e, f, g, h, taken 5 and 5 together. Here n=8, p=5, n−p+1=4; hence the above formula which expresses the number of the permutations of n quantities taken all together. EXAMPLE. Required the number of the permutations of the eight letters a, b, c, d, e, f, g, h. Here n= =8; hence the above formula (2) in this case becomes the number required. 1.2.3.4.5.6.7.8=40320, 202. The number of the permutations of n quantities, supposing them all different from each other, we have found to be 1.2.3... ... (n−1)n. But if the same quantity be repeated a certain number of times, then it is manifest that a certain number of the above permutations will become identical. Thus, if one of the quantities be repeated a times, the number of identical permutations will be represented by 1.2.3............a; and hence, in order to Many writers on algebra confine the term permutations to this class where the quantities are taken all together, and give the title of arrangements or variations to the groups of the n quantities when taken two and two, three and three, four and four, &c. The introduction of these additional designations appears unnecessary; but, in using the word permutations absolutely, we must always be understood to mean those represented by for mula (2), unless the contrary be specified. obtain the number of permutations different from each other, we must divide (2) by 1.2.3............a, and it will then become 1.2.3...............(n − 1)n 1.2.3...............a If one of the quantities be repeated a times, and another of the quantities be repeated ẞ times, then we must divide by 1.2............a × 1.2................................ß; and, in general, if among the n quantities there be a of one kind, ẞ of another kind, y of another kind, and so on, the expression for the number of the permutations different from each other of these n quantities will be Required the numbers of the permutations of the letters in the word algebra. Here n=7, and the letter a is repeated twice; hence formula (3) becomes Required the number of the permutations of the letters in the word caifacarataddarada. Here n=18, a is repeated eight times, c twice, d thrice, r twice; hence the number sought will be 1.2.3.4.5.6.7.8.9.10.11.12.13.14.15.16.17.18 1.2.3.4.5.6.7.8x1.2x1.2.3x1.2 EXAMPLE III. =6616209600. Required the number of the permutations of the product a by c2, written at full length. Here n=x+y+z, the letter a is repeated x times, the letter b, y times, and the letter c, z times; the expression sought will, therefore, be 203. The Combinations of any number of quantities signify the different collections which may be formed of these quantities, without regard to the order in which they are arranged in each collection., Each combination must, therefore, have one letter different from any other of the combinations. Thus the quantities a, b, c, when taken all together, will form only one combination, 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, To find the number of the combinations of n quantities, taken p and p together. Each of these combinations of p quantities being separately permutated, will furnish 1.2.3...p permutations, which, multiplied by the whole number of combinations, will give the whole number of permutations of n quantities, taken * Where numerical or literal factors are combined, the term combination may be considered as signifying the same as product. p and p. Therefore the latter, namely, the whole number of permutations, or n(n-1)(n-2)..........(n−p+1), divided by the number of permutations of each combination, or 1.2.3...p, will give the number of combinations of n quantities, taken P and p. Denoting it by C, we have 204. 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 P and P, are represented by .. (nPp) The number of the permutations of n quantities, taken all together, are represented by ... (nPn) The number of the combinations of n quantities, taken p and p, are represented by . . . (nCp) and so on. It is manifest that the above proposition may be expressed according to this notation by (nPp). (nCp)=(pPp) M. Cauchy employs the notation (m), to express the number of combinations of m letters, taken n at a time. The German notation for the same is When the series of natural numbers, or the letters of the alphabet up to any required number, are to be permuted or combined, an abbreviated notation has been employed as follows: P(1, 2, 3) stands for 123, 132, 213, 231, 312, 321. 2 P(1..4) stands for 12, 13, 14, 21, 23, 24, 31, 32, 34, 41, 42, 43. 3 C(a...e) stands for abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde. If one or more of the numbers or letters may be repeated, this can also be expressed in the notation. Thus, P(1, 1, 2)=112, 121, 211. 2 P(1, 1, 2, 3)=11, 12, 13, 21, 23, 31, 32. 3 C(1, 1, 2, 2, 3)=112, 113, 122, 123, 223. If all the letters, numbers, or single things may be repeated an equal number of times, this can be expressed with the aid of an exponent; thus, Ĉ(1, 2, 3)5, P(0, 1, 2)2, Ĉ(1..7)". 205. If n single things be arranged in combinations of k, or of n-k=r, the number of combinations in either case will be the same, i. e., k C= _n(n−1)... (n−k+1) _ ¿_n(n−1)... (n−r+1) for every new combination of k letters must leave a new one of r letters. By a similar reasoning, if n be divided into three parts, the first k, the second r, and the third s, it may be shown that 206. Cases may occur in which not all possible combinations, but only such as fulfill certain conditions, are required. Many such may be imagined. For instance, where the numbers to be combined increase by a common difference, or by a common ratio, as 1357, 2468, or 124, or 248. The most useful case is where the number in each combination must amount to the same sum. The method of proceeding in this case is to fill up all the places except the last with the lowest numbers, the last place being occupied by the supplementary number necessary to produce the given sum; then diminishing the last number and increasing one of the preceding by the same amount, taking care not to allow a lower ever to follow a higher number. We give examples of such k combinations, the general formula for which is C(1..........n). 3 (1) 10Ĉ(1...7)=127, 136, 145, 235. (2) C(1...8)=1238, 1247, 1256, 1346, 2345. (3) ¿°C(0..5)n=0005, 0014, 0023, 0113, 0122, 1112. (4) 20Ĉ(3........)n=33338, 33347, 33356, 33446, 33455, 34445, 44444. It is easy to be perceived that in two cases this kind of combination is impossible. 1°. When the highest form does not amount to the required sum; and, 2o. When the lowest form exceeds it, as in 207. Similar conditions may be imposed upon permutations. In order that the permutations of a given series of numbers, taken a certain number at a time, should amount always to a given sum, the same rule will apply, with this difference, that lower numbers may follow higher; in other words, the combinations formed by the previous rule may each be permuted. The following examples will render this more intelligible: (1) P(1..8)=18, 27, 36, 45, 54, 63, 72, 81. 3 (2) P(1...)=124, 142, 214, 241, 412, 421. (3) 6P(1.....)n=1113, 1122, 1131, 1212, 1221, 1311, 2112, 2121, 2211, 3111. (4) 4P(0..)n=013, 022, 031, 103, 112, 121, 130, 202, 211, 220, 301, 310. Under this head, also, two contradictory cases occur: 1°. When the highest form amounts to too little; and, 2o. When the lowest form amounts to too much. As, for instance, in P(1... 品は 208. The applications of the theory of permutations and combinations are numerous. One of the most useful is the determination of the coefficients of a series of the form a+bx+cx2+dr3+ex'+...+kx"...,* especially the coefficients of the binomial formula, the method of determining which, by the theory of permutations and combinations, will be given hereafter. Another extensive application of the theory of permutations and combina *These coefficients are supposed to depend upon some given law. A common case is when the number of factors combined in each coefficient is indicated by the exponent of the letter of arrangement, x. tions is to be found in geometric relations, such as where the combinations of a certain number of points, lines, angles, &c., from among a given number of these, are required. Not less useful is this theory in natural science: as in crystalography, when the manifold forms of crystals are required; in chemistry, when the various combinations of chemical elements; and in music, of consonant tones, &c. But perhaps its most important use is in the doctrine of chances, or, as it is mathematically named, the CALCULUS OF PROBABILITIES. The outlines of this extensive subject we shall here briefly indicate, referring the student for further information to the admirable treatises of La Place and Lacroix, and to the practical work of De Morgan. I. Let there be among m possible cases g, which, as fulfilling certain requisitions, are considered as favorable, (m—g)=a unfavorable. Then the ratio of the favorable to all possible cases is called the mathematical probability for the occurrence of a favorable case. The ratio of the unfavorable to all possible cases is the mathematical improbability of the occurrence. If the first be expressed by w, the second by v, then g m น m . (I.) The probability is, therefore, the less, the smaller the number of the favorable in comparison with that of all possible cases, and vice versá. Should all possible cases be favorable, then w=1, which is, therefore, the expression for certainty. Thus the mathematical probability and improbability of a pictured card, of which there are 12, being drawn from 52, are expressed by 40 10 12 3 w=52 13' v=52=13' II. Let there be among m possible cases g favorable, of different (first, second, third, &c.) kinds, expressed by g1, 82, 83, &c., the partial probabilities by w1, w2, w3, &c.; then w=w1+w2+w3+, &c.,=' 81+82 +83+, &c. m that is, the probability of one of several different kinds is equal to the sum of their partial probabilities. Thus, for the probability of one of the six faces of a die, marked 1, 2, or 3, being thrown, we have III. Let the occurrence be favorable only on the supposition that two or more of the single favorable cases concur, then the formula for the compound probability is in which m1, m2, m3, &c., express the possible cases of the partial occurren |