336 LAMBERT. 



consider what is the chance that the predictions will be verified 

 supposing the predictions to be thrown out at random. 



The problem which he is thus led to discuss is really the old 

 problem of the game of Treize, though Lambert does not give this 

 name to it, or cite any preceding writers except Euler's memoir of 

 1751 : see Arts. 162, 280, 430. 



627. We may put the problem thus : suppose n letters to be 

 written and n corresponding envelopes to be directed ; the letters 

 are put at random into the envelopes : required the chance that 

 all, or any assigned number, of the letters are placed in the wrong 

 envelopes. 



The total number of ways in which the letters can be put into 

 the envelopes is n. There is only one way in which all can be 



placed in the right envelopes. There is no way in which just one 

 letter is in the wrong envelope. Let us consider the number of 

 ways in which just two letters are in the wrong envelopes : take 



a pair of letters ; this can be done in — ^- — ^-^ ways j then find 



in how many ways this pair can be put in the wrong envelopes 



without disturbing the others : this can only be done in one way. 



Next consider in how many ways just three letters can be put in 



the wrong envelopes ; take a triad of letters ; this can be done 



7X in 1 ) in 2 ) 



in -^ L \ ways, and the selected triad can be put in 



wrong envelopes in 2 ways, as will be seen on trial. 

 Proceeding thus we obtain the following result, 



A A ^ ^ (*^ — 1) 



A ^ (^ - 1) (^ - ^) A \jl 



1.2.3 



where A^ expresses the number of ways in which r letters, for 

 which there are r appropriate envelopes, can all be placed in wrong 

 envelopes. And 



Ao = ^, A^ = \), A^=l, A^ = 2, .., 



Now Aq, A^, A^, ... are independent of 7i ; thus we can deter- 

 mine them by putting for w in succession the values 1, 2, 3, ... in 



