1 1 2 MONTMORT. 



We must now determine (1) the whole number of possible 

 cases, and (2) the whole number of cases in which the player is 

 arrested at the very beginning. 



(1) We may suppose that 2n cards are to be put in 2n 

 places, and thus [ 27i will be the whole number of possible cases. 



(2) Here we may find the number of cases by supposing that 

 the n upper places are first filled and then the n lower places. 

 We may put m the first place any card oat of the 2/2, then in the 

 second place any card of the 2n — 2 which remain by rejecting the 

 companion card to that we put in the first place, then in the third 

 place any card of the 2n — 4< which remain by rejecting the two 

 companion cards, and so on. Thus the n upper places can be 

 filled in 2" [n ways. Then the n lower places can be filled in [n 

 ways. Hence we get 2*" 1^2 [ji cases in which the player is arrested 

 at the very beginning. 



We may divide each of these expressions by \n if we please 



to disregard the different order in which the n lots may be sup- 



\2n 

 posed to be arranged. Thus the results become M^ and 2" [n 



respectively ; we shall use these forms. 



Let u^ denote the whole number of unfavourable cases, and let 

 /,. denote the whole number of favourable cases when the cards 

 consist of r pairs. Then 



u^=^r[n + t -—^ £ \n-r 2""'', 



the summation extending from r = 2 to r = w — 1, both inclusive. 



For, as we have stated, the player loses by being left with some 

 number of lots, all unbroken, in which the exposed cards contain 

 no pairs. Suppose he is left with n — r lots, so that he has got rid 



\ 71 



of r lots of the original n lots. The factor != g-ives the num- 



\r n — r 



ber of ways in which r pairs can be selected from n pairs ; the 

 factor fi gives the number of ways in which these pairs can be so 

 arranged as to enable the player to get rid of them ; the fiictor 

 \n — r 2""'" gives the number of ways in which the remaining n — r 



pairs can be distributed into n — r lots without a single pair occur- 

 ring among the exposed cards. 



