COMBINATIONS AND PERMUTATIONS. 203 



the extraordinary magnitude of the numbers with which 

 we deal in this subject. Tacquet calculated d that the 

 twenty-four letters of the alphabet may be arranged in 

 more than 620 thousand trillions of orders ; and Schottus 

 estimated e 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 could not have ac- 

 complished 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. Thus 

 if we were to permutate the letters of the name Ann, 

 according to our previous rule, we should obtain 3x2x1, 

 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 



3 2 T 



therefore be - ' or 3, namely Ann, Nan, Nna. In 



the word utility there are two i's and two 's, in respect 

 of both of which pairs the number of permutations must 

 be halved. Thus we obtain 5 4 3 2 r or I2 6 O as 



I . 2 . I . 2 



the number of permutations. The simple rule evidently 

 is that 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 number 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 



d ' Arithmetics Theoria.' Ed. Amsterd. 1704, p. 517. 

 e Rees' ' Cyclopaedia,' art. Cipher. 



