PROBLEMS OF MODERN SCIENCE 



the odds against the happening of some event, 

 now direct your attention to a very similar type c 

 subject the Theory of Partitions which, especi 

 ally through the work of Major Macmahon, is als 

 one of the most rapidly developing, and at th 

 same time most fundamental, branches of Pur 

 Mathematics properly so called. Let us take on 

 of the simplest possible instances the partitions c 

 the small number 5, or the various ways in whid 

 we can make up 5 by the addition of smalle 

 numbers. They are as follows : 



1, 1, 1, 1, 1 1, 2, 2, o, o 



1, 1, 1, 2, o 1, 4, o, o, o 



1, 1, 3, o, o 5, o, o, o, o 



3, 2, o, o, o 



where the order is not regarded as relevant 

 and the total is 7. In a case like this it i 

 easy to enumerate the partitions by writing 

 them down and counting, but it is the worl 

 of a lifetime to find the number of partition: 

 of, let us say, the number 20,000 by any sucl 

 procedure. We need a theory of partitions, anc 

 Major Macmahon has supplied us with a ver) 

 comprehensive one. But my readers may well 

 at this point, ask me what is the practical value 

 and what can ever be the practical value, of such an 

 investigation into the number of partitions of s 

 very large integer number. It might be compared 

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