160 AN EXAMIKATION OF SOME QUESTIONS 



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

 « — » { (— 1)" + (— 1)* ]• . and his third profit will be " 



[u-v {(_1)- +(_!)*} ] (-1)0. 



the succeeding stakes are easily represented in the same man- 

 ner, and his profits stand thus : 



m(— 1)" 



u{-l)'-v{{-l)''}{-lY 

 u{-l)'-v{ (_1)''+(_1)'}(_1)= 

 u{-lY-v{ (-ir+l-lf + (_!/} (-1)" 



&c. &c. 

 the value of the part multiplying u, or 



(-ir+(-i)'+(-ir+&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') b' + (a' + b') c' + {a + b' + c) d' + (a' + b' + c + d') e +, &c. 



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

 quantities a', 6', c, &c. ; and, moreover, that if these latter are 



the 



