COMBINATIONS AND PERMUTATIONS. 211 



7. It follows that the sum of the numbers in any one 

 line is equal to the sum of those in all the preceding lines 

 diminished by a unit. For the sum of the ?ith line is, as 

 already shewn, 2 n ~ l , and the sum of the first n i lines 

 is 2 n ~" 1 i, or less by a unit. 



This enumeration of the properties of the figurate 

 numbers does not approach completeness ; a considerable, 

 perhaps an unlimited, number of less simple and obvious 

 relations might be traced out. Pascal, after giving many 

 of the properties, exclaims l : ' Mais j'en laisse bien plus 

 que je n'en donne ; c'est une chose etrange combien il est 

 fertile en proprietes ! Chacun peut s y exercer/ The 

 arithmetical triangle may be considered a natural classifi- 

 cation of numbers, exhibiting, in the most complete 

 manner, their evolution and relations in a certain point 

 of view. It is obvious that in an unlimited extension of 

 the triangle, each number will have at least two places. 



Though the properties above explained are highly 

 curious, the greatest value of the triangle arises from the 

 fact that it contains a complete statement of the values 

 of the formula (p. 206), for the number of combinations 

 of m things out of n, for all possible values of m and n. 

 Out of seven things one may be chosen in seven ways, 

 and seven occurs in the eighth line of the second column. 

 The combinations of two things chosen out of seven 



*7 V 6 



are - or 21, which is the third number in the eighth 



line. The combinations of three things out of seven are 



3 or 35, which appears fourth in the eighth line. 



In a similar manner, in the fifth, sixth, seventh, and eighth 

 columns of the eighth line I find it stated in how many 

 ways I can select combinations of 4, 5, 6, and 7 things 

 out of 7. Proceeding to the ninth line, I find in succession 



1 ' (Euvres Completes,' vol. iii. p. 251. 

 1> 2 



