92 MONTMORT. 



that is \n-l -2\n-2+\n-S. The number of cases in 



which d is fourth, but neither a, b, nor c in its proper place is 

 \n-l -2\n-2 + \n-S -hn-2-2 \n-S + | n - 4 1, that is 



1/1 — 1 — 3 \n — 2 + 3 \n — S — \n — 4*. And generally the number 



of cases in which the m^^ card is in its proper place, while none 

 of its predecessors is in its proper place, is 



\n-l - (m - 1) 1 71-2 + ^ -^ ^ \n-S 



{m -1) (m- 2) (m-3) 



wt-l 



w — m. 



^ ,71-4 + + (-1) 



"We may supply a step here in the process of Nicolas Bernoulli, 

 by shewing the truth of this result by induction. Let -v/r (771, n) 

 denote the number of cases in which the m"' card is the first that 

 occurs in its right place ; we have to trace the connexion between 

 ^jr (m, n) and yjr {m + 1, n). The number of cases in which the 

 {m + l)**^ card is in its right place while none of the cards between 

 h and the W2*'^ card, both inclusive, is in its right place, is '^^ (m, n). 

 From this number we must reject all those cases in which a is in its 

 right place, and thus we shall obtain yjr {in + 1, n). The cases to 

 be rejected are in number '^ {m, n — 1). Thus 



y^ (in + 1, w) = i/r {in, n) — yfr {in, n — 1). 



Hence we can shew that the form assigned by Nicolas Bernoulli 

 to -^/r (m, n) is universally true. 



Thus if a person undertakes that the m*^ card shall be the first 

 that is in its right place, the number of cases favourable to him is 



'^ (m, n), and therefore his chance is . ' — - , 



\n 



If he undertakes that at least one card shall be in its right 



place, we obtain the number of favourable cases by summing 



^jr (m, n) for all values of m from 1 to n both inclusive : the chance 



is found by dividing this sum by [n. 



Hence we shall obtain for the chance that at least one card is 

 in its right place, 



i_l+i_l, , (- 1)- 



2 [3 li^'"^ \n ' 



