PASCAL. 29 



Take an Arithmetical Triangle with r numbers in its base; 

 then the sum of the numbers in the _29"' horizontal row is equal to 

 the multitude of the combinations of r things taken p at a time. 

 For example, in Art 22 we have a triangle with 10 numbers in 

 its base ; now take the numbers in the 8th horizontal column ; 

 their sum is 1 4-8 + 36, that is 45; and there are 45 combinations 

 of 10 things taken 8 at a time. Pascal's proof is inductive. It 

 may be observed that multitudo is Pascal's word in tlie Latin of 

 his treatise, and multitude in the French version of a part of the 

 treatise which is given in pages 22 — 30 of the volume. 



From this he deduces various inferences such as the followino-. 

 Let there be n things ; the sum of the multitude of the combinations 

 which can be formed, one at a time, two at a time,... , up to n at 

 a time, is 2''— 1. 



At the end Pascal considers this problem. Datis duobus numeris 

 inaequalibus, invenire quot modis minor in majore combinetur. 

 And from his Arithmetical THangle he deduces in effect the follow- 

 ing result ; the number of combinations of r things taken p at 

 a time is 



(^+1) (p + 2) (;; + 3)...r 



■P 



After this problem Pascal adds. 



Hoc problemate tractatum liiiuc absolvere constitiieram, non tamen 

 omniiio sine molestia, cum niulta alia parata liabeam ; sed ubi tanta 

 ubertas, vi moderanda eat fames : his ergo pauca hsec subjiciam. 



Eruditissimus ac milii charisimus, D.D. de Ganieres, circa combina- 

 tiones, assiduo ac peiiitili labore, more suo, incumbens, ac indigens 

 facili constructione ad inveniendum quoties numerus datus in alio dato 

 combinetur, hanc ipse sibi praxim instituit. 



Pascal then gives the rule ; it amounts to this ; the num- 

 ber of combinations of r things taken |) at a time is 



r (>'- 1)... {r-p+ 1) 



■ {p ■ 



This is the form with which we are now most familiar. It 

 may be immediately shewn to agree with the form given before 

 by Pascal, by cancelling or introducing factors into both numerator 

 and denominator. Pascal however savs, Excellentem hanc solu- 



