202 THE PRINCIPLES OF SCIENCE. 



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 of2x2x2x2x we have 



2x3x4x5x 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 permu- 

 tation 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 |_^. I give 

 below the factorials up to that of fifteen : 



6 = 1.2.3 

 24 = i . 2 . 3 . 4 



I 2O = I . 2 .... 5 



720 = 1.2 6 



5,040 = u_ 

 40,320 = IJL 



362,880 = \9_ 



3,628,800 = |io 



39,916,800 = | if. 



479,001,600 = | 



6,227,O2O,8OO = |3 



87,178,291,200 = |4 



1,307,674,368,000 = |5 



The factorials up to (36 are given in Rees' ' Cyclopaedia./ 

 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 

 1*65 would require 529 places of figures. 



Many writers have from time to time remarked upon 



