3 .2. I
I.2

1.2. I. 2

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 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 numbers of permutations must be halved. Thus we obtain 7.6.5.4.3.2.1 or 1260, as the number of permutations. The simple rule evidently is 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 numbers 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 letters, of which four are '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, which is not the one4 X 4 X 2 thousandth part of what we should obtain were all the letters different.

II

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 a group of 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 x 25 x 24. 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. To get the number of combinations, then, we must divide the whole number of ways of choosing, by six, the number of permutations of three things, 26 X 25 X 24 -or 2,600.

obtaining

I X 2 X 3

It is apparent that we need the doctrine of combinations 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 x 27 x 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

8.7

is possible in or 28 ways; of 3 out of 8 in 1.2

8.7.6.5
1.2.3.4

8.7.6 1.2.3

or 56 ways; of 4 out of 8 in or 70 ways; and it may be similarly shown that for 5, 6, 7, and 8 planets, meeting at one time, the numbers of ways are 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

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 formula of mathematical analysis traces of its influence may 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 2 "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

1 Euvres Complètes de Pascal (1865), vol. iii. p. 302. Montucla states the name as De Gruières, Histoire des Mathématiques, vol. iii. p. 389. 2 Histoire des Mathématiques, vol. iii. p. 378.

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 Bernoulli to demonstrate fully the importance of the figurate numbers, as they are also called. In his treatise De Arte Conjectandi, he points out their application in the theory of combinations and probabilities, and remarks of the Arithmetical Triangle, "It not only contains the clue to the mysterious doctrine of combinations, but it is also the ground or foundation of most of the important and abstruse discoveries that have been made in the other branches of the mathematics." 1

The numbers of the triangle can be calculated in a very easy manner by successive additions. We commence with unity at the apex; in the next line we place a second unit to the right of this; to obtain the third line of figures we move the previous line one place to the right, and add them to the same figures as they were before removal; we can then repeat the same process ad infinitum. The fourth line of figures, for instance, contains 1, 3, 3, ; moving them one place and adding as directed we obtain :

Carrying out this simple process through ten more steps we obtain the first seventeen lines of the Arithmetical Triangle as printed on the next page. Theoretically speaking the Triangle must be regarded as infinite in extent, but the numbers increase so rapidly that it soon becomes impracticable to continue the table. The longest table of the numbers which I have found is in Fortia's "Traité des Progressions" (p. 80), where they are given up to the fortieth line and the ninth column.

1 Bernoulli, De Arte Conjectandi, translated by Francis Maseres. London, 1795, P. 75.

THE ARITHMETICAL TRIANGLE.

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I

I

3456789

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