18 PASCAL AND FERMAT. 



23. Pascal applies liis Arithmetical Triangle to various subjects ; 

 among tliese we have the Problem of Points, the Theory of Com- 

 binations, and the Powers of Binomial Quantities. We are here 

 only concerned with the application to the first subject. 



In the Arithmetical T^^iangle a line drawn so as to cut off 

 an equal number of units from the top horizontal row and the 

 extreme left-hand vertical column is called a base. 



The bases are numbered, beginning from the top left-hand 

 corner. Thus the tenth base is a line drawn through the num- 

 bers 1, 9, 36, 84, 12G, 12G, 84, 36, 9, 1. It will be perceived that 

 the r*^ base contains r numbers. 



Suppose then that A wants m points and that B wants n 

 points. Take the {m + ii)^^ base; the chance oi A is to the chance 

 of B as the sum of the first n numbers of the base, beginning at 

 the highest row, is to the sum of the last m numbers. Pascal 

 establishes this by induction. 



Pascal's result may be easily she^vn to coincide with that 

 obtained by other methods. For the terms in the (m + ti)"^ base 

 are the coefficients in the expansion of (1 -f xY'^''~^ by the Binomial 

 Theorem. Let m + n — l=r\ then Pascal's result amounts to 

 saying that the chance of A is proportional to 



- r (r — 1) r (r — l) ... (r — n-\-2) 



I . z n — 1 



and the chance of B proportional to 



Ij^yj^ r (r-1) ^ ^^^^ ^ r{r-l)...{r- m + 2) 



1.2 ^^1-1 



This agrees with the result now usually given in elementary 

 treatises; see Algebra, Chapter Liii. 



24. Pascal then notices some particular examples. (1) Sup- 

 pose that A wants one point and B wants n points. (2) Suppose 

 that A wants n — 1 points and B wants n points. (3) Suppose 

 that A wants n— 2 points and B wants n points. An interesting 

 relation holds between the second and third examples, which we 

 will exhibit. 



