different permutations according as A stands first, second, or third; having decided the place of A, there are two places between which we may choose for B; and then there remains but one place for C. Accordingly the permutations of these letters will be altogether 3 × 2 × 1 or 6 in number. With four things or letters, A, B, C, D, we shall have four choices of place for the first letter, three for the second, two for the third, and one for the fourth, so that there will be altogether, 4 × 3 × 2 × 1, or 24 permutations. The same simple rule applies in all cases; beginning with the whole number of things we multiply at each step by a number decreased by a unit. In general language, if n be the number of things in a combination, the number of permutations is
If we were to re-arrange the names of the days of the week, the possible arrangements out of which we should have to choose the new order, would be no less than 7 . 6 . 5 . 4 . 3 . 2 . 1, or 5040, or, excluding the existing order, 5039.
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, but increases the permutations by a factor continually growing. Instead of 2 × 2 × 2 × 2 × .... we have 2 × 3 × 4 × 5 × .... and the products of the latter expression immensely exceed those of the former. These products of increasing factors are frequently 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 n!. I give below the factorials up to that of twelve:—
| 24 = | 1 . 2 . 3 . 4 |
|---|---|
| 120 = | 1 . 2 . . . 5 |
| 720 = | 1 . 2 . . . 6 |
| 5,040 = | 7! |
| 40,320 = | 8! |
| 362,880 = | 9! |
| 3,628,800 = | 10! |
| 39,916,800 = | 11! |
| 479,001,600 = | 12! |
The factorials up to 36! are given in Rees’s ‘Cyclopædia,’ art. Cipher, and the logarithms of factorials up to 265! are to be found 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 calculated4 that the twenty-four [sic] letters of the alphabet may be arranged in more than 620 thousand trillions of orders; and Schott estimated5 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 would 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.