438 Wisconsin Academy of Sciences, Arts, and Letters. 



Lastly, if we divide by the total number of combinations, w» 

 have the 'probability of the error 



52771— (a:i+a,-f.... 4-0'^)?;' yiz: 



%t{y—l)\a^a.^...a\^^^ —'^{m^s^a) ^^[m—e^af —...([5] 



Therefore the probability of any particular error approaches 

 zero as n is increased without limit. 



§ 5. 



If it is required to find the formula which expresses thb 

 number of combinations within certain limits, it is only neces- 

 sary to choose a series of exponents in tlie product of the series 

 [3] of §4 and sum the coellicients. For this purpose let u» 

 return to [4] and take the cocfiicient of the term z^ , 



{y-\)\ I 



^ t^ ^ —^{f-SiUf ^ -\-:S{t—82U) 



.-!_ ) 



• • • • g 



and for t vv'ritc in order all its values from t = 1 to t = t and 

 sum them, which will give the expression required. First t^—l 

 is changed into the series 



1+2^-1 ^.s"-! +4"-"^ ^,,„^r-l 5 



this series is equivalent to 



..i(i)'-+(i)'-Vj3)'-V...H.(4)-|| 



which, if t is assumed infinite, can be expressed by the integral 



Jo 



