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 3 x 2 x i, 

 or 6 orders ; but half of tliese arrangenients would be 

 identical with the other half, because the interchange of 

 tlie two ns has no effect. The really different orders will 



-2 2 1 



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



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

 of both of which pairs the numbers of permutations must 



be halved. Thus we obtain — — . s • ^ • i • - • q^. J260, as 



I . 2 . I . 2 



the number of permutations. The simple rule evidently 



i;s — wlien some things or letters are undistinguished, 



proceed in the first place to calculate all the possible 



])ermutations as if all were different, and then divide by 



the numbers of possible permutations of those series of 



things which are not distinsiuislied, and of which the 



permutations have therefore been counted in excess. 



Tlius since the word Utilitarianism contains fourteen 



letters, of which four are i&, two a's, and two ^'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 wq can get in like manner 



III , 1 • 1 



, = r- or 34,6 i;o permutations, which is not the one- 



1 4 X |_4 X |_2 J^' -> 1 



thousandth part of what we should obtain were all the 

 letters different. 



Calculation of Numher of Comhinations. 



Although in many questions both of art and science 

 we need to calculate tlie number of permutations on 

 account of their own interest, it far more frequently 

 happens in scientific subjt^cts that the}' possess but an 

 indirect interest. As I liave already pointed out, we 

 almost always deal in the logical and mathematical 

 sciences with comhinations, and variety of order enters 



