BEGUELIN. 829 



All this is well known, as Beguelin says, but it is introduced 

 by him as leading the way for his further investigations. 



606. Such cases as a, a, a, a, a cannot occur in the lottery 

 because no number is there repeated. Let the second file be 

 raised one letter, the third file two letters; and so on. Thus 

 we have 



a h c d e 



h c d e f 



m 



We have thus 13 — 4 complete files, that is 9 complete files ; 

 and, proceeding as before, the number of associations is found to be 



9 X 10 X 11 X 12 X 13 ,. , , . 



— — ^ — -. z — ; that IS, the number is what we know to 



1x2x3x4x0 



be the number of the combinations of 13 things taken 5 at a time. 



607. Suppose now that we wish to find the number of asso- 

 ciations in which there is no sequence at all. Raise each file two 

 letters instead of one, so that we now have 



