246 MISCELLANEOUS PROBLEMS [CH. XI 



cards; and what permutations will give a certain number of 

 hits in a certain order. 



Cay ley* showed that there are 9 arrangements of a pack 

 of four cards in which no hit will be made, 7 arrangements in 

 which only one hit will be made, 3 arrangements in which only 

 two hits will be made, and 5 arrangements in which four hits 

 will be made. 



Prof. Steen-f has investigated the general theory for a pack 

 of n cards. He has shown how to determine the number of 

 arrangements in which x is the first hit [Arts. 3 — 5]; the 

 number of arrangements in which 1 is the first hit and x is the 

 second hit [Art. 6] ; and the number of arrangements in which 

 » 2 is the first hit and x the second hit [Arts. 7 — 8] ; but beyond 

 this point the theory has not been carried. It is obvious that, 

 if there are n — 1 hits, the nth hit will necessarily follow. 



The French game of treize is very similar. It is played 

 with a full pack of fifty-two cards (knave, queen, and king 

 counting as 11, 12, and 13 respectively). The dealer calls out 

 1, 2, 3, ..., 13, as he deals the 1st, 2nd, 3rd, .... 13th cards 

 respectively. At the beginning of a deal the dealer offers to 

 lay or take certain odds that he will make a hit in the thirteen 

 cards next dealt. 



* Quarterly Journal of Mathematics, 1878, vol. xv, pp. 8 — 10. 

 t Ibid., vol. xv, pp. 230—241. 



