EULER. 251 



what we have just seen ; this therefore is the niunber of ways 

 in which with the given die the ace can come up without the deux. 

 Subtract this number from the number of ways in which the ace 

 can come up with or without the deux, and we have left the 

 number of ways in which the ace can come up luith the deux. 

 Thus the result is 



n^-{n-ir-\{n-\r-{n-^.rr, 

 that is, if -2{n- Vf -f {n - 2f. 



De Moivre in like manner briefly considers the case in wdiich 

 the ace, the deux, and the tray are to come up ; he then states 

 what the result will be when the ace, the deux, the tray, and 

 the quatre are to come up ; and finally, he enunciates verbally 

 the general result. 



De Moivre then proceeds to shew how approximate numerical 

 values may be obtained from the formula ; see Art. 292. 



450. The result may be conveniently expressed in the nota- 

 tion of Finite Differences. 



The number of ways in which m specified faces can come up 

 is A'" (71 — mY ; where m is of course not greater than n. 



It is also obvious that if m be greater than x, the event 

 required is impossible ; and in fact we knoAv that the expression 

 A"' (?z — my vanishes when ?n is greater than x. 



Suppose 71 = m ; then the number of ways may be denoted by 

 A^O"^ ; the expression written at full is 



,f _ ,, (,, _. 1)- a. ^^j-^-^ {^ri-Tf-... 



451. One particular case of the general result at the end 

 of the preceding Article is deserving of notice. If we jDut x = n, 

 we obtain the number of ways in which all the 71 faces come up 

 in n throws. The sum of the series wdien x = 7i is known to be 

 equal to the product 1.2.3...??, as may be shewn in various 

 ways. But we may remark that this result can also be obtained 

 by the Theory of Probability itself; for if all the 7i faces are 

 to appear in ii throws, there must be no repetition ; and thus the 



