COMBINATIONS AND PERMUTATIONS. 221 



various duration and succession, and the scheme would be 

 easy of execution if needed. 



Let us calculate the number of combinations of dif- 

 ferent orders which may arise out of the presence or 

 absence of a single mark, say A. Thus in 



i A i Ai ran i i m i \ i 



we have four distinct varieties. Form them into a group 

 of a higher order, and consider in how many ways we 

 may vary that group by omitting one or more of the 

 component parts. Now, as there are four parts, and any 

 one may be present or absent, the possible varieties will 

 be 2 x 2 x 2 x 2, or 16 in number. Form these into a new 

 whole, and proceed again to create variety by omitting 

 any one or more of the sixteen. The number of pos- 

 sible changes will now be 2.2.2.2.2.2.2.2.2.2.2.2.2.2.2.2, 

 or 2 16 , and we can repeat the process again and again if 

 we wish. It will be easily seen that we are imagining 

 the creation of objects, whose numbers are represented 

 in the series of expressions 



2 

 2 2 



222 



2222, and so on. 

 At the first step we have 2 ; at the next 2 2 , or 4 ; 



2 



at the third 2 2 , or 16, numbers of very moderate amount. 

 Let the reader calculate the next term, and he will 

 be surprised to find it leap up to 65,536. But at the 

 next step he has to calculate the value of 65,536 

 two's multiplied together, and it is so great that we 

 could not possibly compute it, the mere expression of 

 the result requiring 19,729 places of figures. But go 

 one step more and we pass the bounds of all reason. 

 The sixth order of the powers of two becomes so 

 great, that we could not even express the number of 



