1 60 AN EXAMINATION OF SOME QUESTIONS 



if that number is odd. These may be represented thus : 

 u — v i ( — l) a -f- ( — 1) \ » an <l his third profit will be 



[«,-» {(-1)° + (-!)'} ] (-l)'i 

 the succeeding stakes are easily represented in the same man- 

 ner, and his profits stand thus : 



(-i)" 

 .1)*— 1.{ (—1)°} 



(-i)'-v{(-i)'+(-if}{-iy 



i-lf-*{ (-!)*+(- I)' + (-!)'} (-1)' 



.!)«_„{ (_ 1)- + (-1)'+ (- 1)' + (-1)'} (-1)% 

 &c. &c. 

 the value of the part multiplying u, or 



(_ !)* + (_ ])»+(- l) c + &c. 



is easily found, since we know that p of the quantities a, 6, c, 

 are even numbers, and q of them odd numbers. The first 

 part of the expression is therefore 



( p _ q ) u . 



In order to determine the second part, we must observe, 

 that the expression 



(a) V + (a' + 6') c' + (a + 6' + c) <T + (a' + b' + c + d') e +, &c 



is equal to the sum of all the combinations, two by two of the 

 quantities a, b\ c, &c. ; and, moreover, that if these latter are 



the 



