SOG LAGRANGE. 



We must find the coefficient of x>^ in the expansion of 



{^-« + 2a?-"+^ + ... + aa?-' + (a + 1) x° + ao; + .. . + 2a;"-' + a;"}", 



and divide it by the vahie of this expression when aj = 1, which is 

 the whole number of cases ; thus we obtain the required pro- 

 babiHty. 



Now 1+ 2x + 3a;'+ ... + (a+ 1) a;"+ ... + 2a;""-' + x^ 



Jia 



a+i\ 2 



= {iJ,x+x'-\-... + xy=(Kj^^ . 



Hence finally the required probability is the coefficient of 

 a;'* in the expansion of 



1 a;-"" (1 - a;"-^y" ^ 

 that is the coefficient of a;'^"^"" in the expansion of 



^,(i-.r(i-o- 



Lagrange gives a general theorem for effecting expansions, of 

 which this becomes an example ; but it will be sufficient for our 

 purpose to employ the Binomial Theorem. We thus obtain for 

 the coefficient of a?'^"^"" the expression 



+ \ ^ ^ </)(7ia+At + l-2«-2) 



%i {271 - 1) {2n - 2) ^ , . o ON 1 

 12 3 " 0(wa+/^ + l-3a-S)4-...|; 



where (r) stands for the product 



r (r+1) (r + 2) . . . (r + 2w - 2) ; 



the series within the brackets is to continue only so long as r is 

 positive in (j) (r). 



565. We can see a priori that the coefficient of xf^ is equal 

 to the coefficient of x~'^, and therefore when we want the former 

 we may if we please find the latter instead. Thus in the result of 



