The reader will see that the numbers which we reach in questions of permutation, increase in a more extraordinary manner even than in combination. Each new object or term doubles the number of combinations (p. 195), but increases the permutations by a factor continually growing. Instead of 2 x 2 x 2 x 2 x .....

we have

2 × 3 × 4 × 5 × and the products of the latter expression indefinitely exceed those of the former. These products of continually increasing factors are constantly employed, as we shall see, in questions both of permutation and combination. They are technically called factorials, that is to say, the product of all integer numbers, from unity up to any number n, is the factorial of n, and is often indicated symbolically by 2. I give below the factorials up to that of fifteen :

The factorials up to 36 are given in Rees' " 'Cyclopædia,' art. Cipher, and the logarithms of products up to 265 are given at the end of the table of logarithms published under the superintendence of the Society for the Diffusion of Useful Knowledge (p. 215). To express the factorial 265 would require 529 places of figures.

Many writers have from time to time remarked upon

the extraordinary magnitude of the numbers with which we deal in this subject. Tacquet calculated that the twenty-four letters of the alphabet may be arranged in more than 620 thousand trillions of orders; and Schottus estimated that if a thousand millions of men were employed for the same number of years in writing out these arrangements, and each man filled each day forty pages with forty arrangements in each, they could not have accomplished the task, as they would have written only 584 thousand trillions instead of 620 thousand trillions.

In some questions the number of permutations may be restricted and reduced by various conditions. Some things in a group may be undistinguishable from others, so that change of order will produce no difference. Thus if we were to permutate the letters of the name Ann, according to our previous rule, we should obtain 3 × 2 × 1, or 6 orders; but half of these arrangements would be identical with the other half, because the interchange of the two n's has no effect. The really different orders will therefore be 3.2.1 or 3, namely Ann, Nan, Nna. In the word utility there are two i's and two t's, in respect of both of which pairs the number of permutations must be halved. Thus we obtain 7.6.5.4.3.2.1

1.

2

I 2 I 2

or 1260, as

the number of permutations. The simple rule evidently is that when some things or letters are undistinguished, proceed in the first place to calculate all the possible permutations as if all were different, and then divide by the number of possible permutations of those series of things which are not distinguished, and of which the permutations have therefore been counted in excess. Thus since the word Utilitarianism contains fourteen

d'Arithmeticæ Theoria.' Ed. Amsterd. 1704, P. 517.
e Rees' Cyclopædia,' art. Cipher.

letters, of which four are i's, two a's, and two t's, the number of distinct arrangements will be found by dividing the factorial of 14, by the factorials of 4, 2, and 2, the result being 908,107,200. From the letters of the word Mississippi we can get in like manner or 34,650 permutations, or not one-thousandth part of what we should obtain were all the letters different.

II

4 X 4 X 2

Calculation of Number of Combinations.

Although in many questions both of art and science we need to calculate the number of permutations on account of their own interest, it far more frequently happens in scientific subjects that they possess but an indirect interest. As I have already pointed out, we almost always deal in the logical and mathematical sciences with combinations, and variety of order enters only through the inherent imperfections of our symbols and modes of calculation. Signs must be used in some order, and we must withdraw our attention from this order before the signs correctly represent the relations of things which exist neither before nor after each other. Now, it often happens that we cannot choose all the combinations of things, without first choosing them subject to the accidental variety of order, and we must then divide by the number of possible variations of order, that we may get to the true number of pure combinations.

Suppose that we wish to determine the number of ways in which we can select three letters out of the alphabet, without allowing the same letter to be repeated. At the first choice we can take any one of 26 letters; at the next step there remain 25 letters, any one of which may be joined with that already taken; at the third step there will be 24 choices, so that apparently the whole

number of ways of choosing is 26 × 25 × 24. x x But the fact that one choice succeeded another has caused us to obtain the same combinations of letters in different orders; we should get, for instance, a, p, r at one time, and p, r, a at another, and every three distinct letters will appear six times over, because three things can be arranged in six permutations. Thus the true number of combinations will be

24 X 23 X 22

IX 2 X 3

or 2024.

It is apparent that we need the doctrine of permutations in order that we may in many questions counteract the exaggerating effect of successive selection. If out of a senate of 30 persons we have to choose a committee of 5, we may choose any of 30 first, any of 29 next, and so on, in fact there will be 30 x 29 x 28 × 27 × 26 selections; but as the actual character of the members of the committee will not be affected by the accidental order of their selection, we divide by 1 × 2 × 3 × 4 × 5, and the possible number of different committees will be 142,506. Similarly if we want to calculate the number of ways in which the eight major planets may come into conjunction, it is evident that they may meet either two at a time or three at a time, or four or more at a time, and as nothing is said as to the relative order or place in the conjunction, we require the number of combinations. Now a selection of 2 out of 8

or 56 ways; of 4 out of 8 in

of 3 out of 8 in

8.7.6.5

1.2.3.4

8.7.6

1.2.3

or 70 ways; and it may be similarly shown that for 5, 6, 7, and 8 planets, meeting at one time, the number of ways is 56, 28, 8 and I. Thus we have solved the whole question of the variety of conjunctions of eight planets; and adding all the numbers together, we find that 247 is the utmost possible number of modes of meeting.

In general algebraic language, we may say that a group

of m things may be chosen out of a total number of n things, in a number of combinations denoted by the formula n. (n−1) (n−2) (n−3). ... (n−m + 1)

The extreme importance and significance of this formula seems to have been first adequately recognised by Pascal, although its discovery is attributed by him to a friend, M. de Ganières. We shall find it perpetually recurring in questions both of combinations and probability, and throughout the formulæ of mathematical analysis traces of its influence will be noticed.

The Arithmetical Triangle.

The Arithmetical Triangle is a name long since given to a series of remarkable numbers connected with the subject we are treating. According to Montucla "this triangle is in the theory of combinations and changes of order, almost what the table of Pythagoras is in ordinary arithmetic, that is to say, it places at once under the eyes, the numbers required in a multitude of cases of this theory.' As early as 1544 Stifels had noticed the remarkable properties of these numbers and the mode of their evolution. Briggs, the inventor of the common system of logarithms, was so struck with their importance that he called them the Abacus Panchrestus. Pascal, however, was the first who wrote a distinct treatise on these numbers, and gave them the name by which they are still known. But Pascal did not by any means exhaust the subject, and it remained for James Bernouilli to demonstrate fully the importance of the figurate numbers, as they are also called. In his treatise De Arte Conjectandi, he points out their applif 'Euvres Complètes de Pascal' (1865), vol. iii. p. 302. Montucla states the name as De Gruières, 'Histoire des Mathématiques,' vol. iii. p. 389. 8 Histoire des Mathématiques,' vol. iii. p. 387.

h Leslie, 'Dissertation on the Progress of Mathematical and Physical Science,' Encyclopædia Britannica.