560 355 which is the proportion assigned by Metius. Ex. 4. The mean tropical year consists of 3 65-2422642 days. The fraction 2422642, reduced to a continued fraction, gives as successive convergents 1 7 8 39 47 321 4 1 * S, 29 33 161 194 1325 To make the civil year approximate to the tropical, 1 leap year in 4 (the Julian Calendar) serves but imperfectly. 7 leap years in 29 would be inconvenient. The Gregorian Calendar, now in use, is based on combining the fractions and - , by doubling the numerator and denominator 194 4 J of the former, and trebling those of the latter, and adding 97 them respectively. The resulting fraction is , giving 97 leap years in 400 years, instead of 100 as the Julian does. This diminution of 3 leap years in 400 years is produced periodically, by causing years which indicate the completion of centuries not to be leap years unless the number of centuries is divisible by 4. Thus, 1900 will not be a leap year. SECT. XVIII. PERMUTATIONS, COMBINATIONS, AND PROBABILITIES. 139. Hither to we have supposed the letters of the alphabet, a, b, c, &c., to stand for arithmetical quantities of some kind or other. Now we have to employ them, as in geometry, to represent magnitudes or objects, such as pens, pencils, &c., and to investigate the numbers of different ways in which a given set of them can be grouped according to a certain law. Permutations are their arrangements in a line, reference being had to the order of sequence ; thus ab and ba are the two permutations of a and b ; combinations are their arrangements in groups, without reference to the order of sequence; thus abc is a combination involving a, b, and c ; and lac is the same combination, both con sisting simply of a, b, and c grouped together. Prop. 1. To find the number of permutations of n things (1), two and two (2), three and three, &c., together. Set aside a, and lay down the other things in a line ; place a before each of them in succession, and you obtain ab, ac, ad, &c., i.e., n- arrangements, each containing two things, with a first. In the same way you can form n - 1 arrangements, each containing two things, with b first. The same is true of each of the other letters, and as there are n of them, the total number of arrangements of the n things, two together, is n(n- 1). Again, lay aside a, and group the other n - 1 things, two together; as we have just shown, there are (re - 1) (n - 2) such groups. Place a before each of them, and there will he formed (n-l)(n-2) arrangements, each containing three things, with a first; and there can be no more arrangements with a first. Treat b, c, &c., in the same manner, and it will appear that there are (n-l)(n-2) groups of things, three to gether, in which every separate thing in succession stands first. Hence, the total number of arrangements, three and three, is n(n 1) (n - 2). By proceeding in the same manner we shall find the total number of permutations of n things, r together, to be (-!) (n-r+l). [PERMUTATIONS, ETC. Cor. The number of permutations of n things, all to gether, is n(n -1)....3.2.1. Prop. 2. To find the number of combinations of n things, together. Let x be the number required. Take any one of the x groups of r things. The number of permutations which can be formed with it will be (Prop. 1. Cor.) r(r-l) 1, or 1 . 2 . . . r . Now, since each of the x groups is different from all the others, if we treat each of the x groups separately in his way, we shall form 1.2 r x x permutations, all different. Also, since the x groups contain every possiblc- combination of the n things, r together, we shall thus have formed all the permutations which can be formed; and conse quently (Prop. 1) the number is n(n - 1) . . . (n - r + 1) , n(n-l) (n-r + l) Prop. 3. To find the number of combinations which can be formed of n sets of things, containing respectively r, s, t, &c. things, by taking one from each set to form a com bination. 1. Let there be two sets, one containing r and the other s things. Any one (say a) of the r things may be placed succes sively with each of the s things, and thus form s groups, in each of which a appears. The same is true of b, c, &c. ; i.e., each of the r things gives rise to s groups, . . the number required is rs. 2. Any one of the t things may be placed in succession with each of the groups of two things referred to in Case 1, so that every one of the t things will give rise to rs com binations of three things ; . . the number required is rst. The same may be indefinitely extended. 140. The first and most obvious application of the theory of combinations is to the doctrine of chances. As, however, this application will form the subject of a separate article, all that is requisite for us now to do is to indicate the connecting link between the two subjects. If we agree to designate certainty by unity, then the chance of an event happening, when it is less than cer tainty, will be designated by a proper fraction. Thus, if the average number of wet days and of dry is the same, the chance of any day named at random turning out wet will be represented by the fraction - ; that is, if the num ber of days under consideration be 100, the chance is 50 number of wet days . . ,. , , . or ,. 1 J Chance is accordingly de- 100 total number of days number of favourable events fined by the fraction r j J total number of events. The only proposition by which chances are combined that we shall offer is this. If there are two events, and the probability of one of them happening to be ^ , and of the other - ; then the pro bability that both will happen is T-: For a and c may be taken to represent the favourable events respectively, and be combined (Art. 139, Prop. 3) so as to give ac ways in which they may happen together. And in the same way b and d may be combined to give the total number of events. Ex. A bag contains 3 white and 4 black balls. Find the chance of drawing (1) two white balls; (2) a white and a black; (3) one white at least, when two balls are drawn. The chance of drawing two white balls is the fraction, Number of combinations of 3 things, 2 together
Number of combinations of 7 things, 2 together.
