COMBINATIONS AND PERMUTATIONS. 201 



proper proportions and well mixed. But this indifference 

 of order does not usually extend to the events of physical 

 science or the operations of art. The change of mechanical 

 energy into heat is not exactly the same as the change 

 from heat into mechanical energy ; thunder does not in- 

 differently precede and follow lightning ; it is a matter of 

 some importance that we load, cap, present, and fire a rifle 

 in this precise order. Time is the condition of all our 

 thoughts, space of all our actions, and therefore both in 

 art and science we are to a great extent concerned with 

 permutations. All language, for instance, treats different 

 permutations of letters as having different meanings. 



Permutations of certain things are far more numerous 

 than combinations of those things, for the obvious reason 

 that each distinct thing is regarded differently according 

 to its place. Thus the letters A, B, C, will make different 

 permutations according as A stands first, second, or third ; 

 having decided the place of A, there are two places 

 between which we may choose for B ; and then there 

 remains but one place for C. Accordingly the permuta- 

 tions of these letters will be altogether 3x2x1 or 6 in 

 number. With four things or letters, A, B, C, D, we 

 shall have four choices of place for the first letter, three 

 for the second, two for the third, and one for the fourth, 

 so that there will be altogether 4x3x2x1, or 24 

 permutations. The same simple rule applies in all cases ; 

 beginning w;th the whole number of things we multiply 

 at each step by a number decreased by a unit. In general 

 language, ifn be the number of things in a combination, the 



number of permutations is n (n i) (n 2) 4.3.2. i . 



Thus, if we were to re-arrange the names of the days of the 

 week, the possible arrangements out of which we should 

 have to choose the new order, would be no less than 

 7.6.5.4.3.2.1, or 5040, or, excluding the existing 

 order, 5039. 



