17-i BE MOIVKE. 



and fill up the remaining places with the letters aaaahlhhh in this 

 order ;- or put a in any one of the last seven places, and fill up the 

 remaining places with the letters aaahbhhhh in this order ; we thus 

 obtain the ten admissible cases. 



The next term is 4t5a%^. There are forty -five ways in which 

 i?'s capital may be exhausted while A wins only five games. 

 For let there be ten places. Take any two of the first three 

 places and put h in each, and fill up the remaining places with 

 the letters aaaaabhh in this order. Or take any two of the 

 last seven places and put a in each, and fill up the remaining 

 places with the letters aaahhhhh in this order. Or put h in any 

 one of the first three places and a in any one of the last seven ; 

 and fill up the remaining places with the letters aaaabhhh in this 

 order. On the whole we shall obtain a number equal to the num- 

 ber of combinations of 10 things taken 2 at a time. The following 

 is the general result : suppose we have to arrange r letters a and 

 s letters h, so that in each arrangement there shall be n more 

 of the letters a than of the letters h before we have gone through 

 the arrangement ; then if r is less than s + n the number of 

 different arrangements is the same as the number of combina- 

 tions of T -\-s things taken r — w at a time. For example, let 

 r = 6, s = 4, w = 3 ; then the number of different arrangements is 



10 X 9 X 8 ,, , . ^^_ 



— — rr . that IS 120. 



1x2x3 



The result which we have here noticed was obtained by Mont- 

 mort, but in a very unsatisfactory manner : see Art. 182. 



De Moivre's first solution of his Problem LXV. is based on the 

 same principles as Montmort's solution of the general problem 

 of the Duration of Play. 



809. De Moivre's second solution of his Problem LXV. con- 

 sists of a formula which he gives without demonstration. Let us 

 return to the expression in Ai't. 306, and suppose m infinite. Then 

 the chance of A for winning precisely at the (n + 2ic)*'' game is 

 the coefficient of f^^"" in the expansion of 



(l + V(i-4c) 



n > 



