IX.] COMBINATIOINS AND PERMUTATIONS. 179 



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 2X2X2X2X we have 2X3X4X5X 



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 ques- 

 tions 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 = I . 2 . 3 . 4 



120 = I . 2. ... 5 



720 = I . 2 .... 6 



5,040= [7. 

 40,320 = ^ 

 362,880 = L9 

 3,628,800 = [ip 

 39,916,800 = [£1 

 479,001,600 = [12 

 . The factorials up to [3^ are given in Eees's ' Cyclopaedia,' 

 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 5 29 places of figures. 



Many writers have from time to time remarked upon 

 the extraordinaiy 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 Schott 

 estimated ^ that if a thousand millions of men were em- 

 ployed 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 writteti only 

 584 tliousand trillions instead of 620 thousand trillions. 



^ Arithmeticcn Theoria. Ed. ^Viiisterd. 1704. p 517. 

 ' liees's Cijclopctdia, art. Ciplicr. 



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