180 THE PRINCIPLES OF SCIENCE. [CHAP 



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 ?i's has no effect. The really different orders will 



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



the word utility there are two i's and two t's, in respect 

 of both of which pairs the numbers of permutations must 



be halved. Thus we obtain ' 5 ^ 4 ' 3 ^ 2 ' ' or 1260, as 



the number of permutations. The simple rule evidently 

 is 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 numbers 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 

 letters, of which four are z's, two a's, and two t's, the 

 number of distinct arrangements will be found by 

 dividing the factorial of 14, by the factorials of 4, 2, 

 and 2, the result being 908,107,200. From the letters 

 of the word Mississippi we can get in like manner 



i ^7~\ ^TT~ or 34^5 permutations, which is not the one 

 1 4 X I 4 X {^2 



thousandth part of what we should obtain were all the 

 letters different. 



Calculation of Number of Combinations. 



Although in many questions both of art and science 

 we need to calculate the number of permutations on 

 account of their own interest, it far more frequently 

 happens in scientific subjects that the)' possess but an 

 indirect interest. As I have already pointed out, we 

 almost always deal in the logical and mathematical 

 sciences with combinations, and variety of order enters 



