204 . THE PRINCIPLES OF SCIENCE. 



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



, ' , or 34,6 50 permutations, or not one-thousandth 



[X|j^X|_2_ 



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 they 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 

 only through the inherent imperfections of our symbols 

 and modes of calculation. Signs must be used in some 

 order, and we must withdraw our attention from this order 

 before the signs correctly represent the relations of things 

 which exist neither before nor after each other. Now, it 

 often happens that we cannot choose all the combinations 

 of things, without first choosing them subject to the 

 accidental variety of order, and we must then divide by 

 the number of possible variations of order, that we may 

 get to the true number of pure combinations. 



Suppose that we wish to determine the number of 

 ways in which we can select three letters out of the 

 alphabet, without allowing the same letter to be repeated. 

 At the first choice we can take any one of 26 letters ; at 

 the next step there remain 25 letters, any one of which 

 may be joined with that already taken ; at the third step 

 there will be 24 choices, so that apparently the whole 



