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 calculated[97] that the twenty-four [sic] letters of the alphabet may be arranged in more than 620 thousand trillions of orders; and Schott estimated[98] 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. [97] *Arithmeticæ Theoria.* Ed. Amsterd. 1704. p. 517. [98] Rees’s *Cyclopædia*, art. *Cipher*.