MONTMORT. 83 



three faces will be black and one white, six ways in which two 

 faces will be black and two white; and so on. Then he reasons 

 thus: we know by the rules for multiplication that in order to 

 raise a + h to the fourth power (1) we must take the fourth power 

 of a and the fourth power of h, which is the same thing as taking 

 the four black faces and the four white faces, (2) we must take 

 the cube of a with b, and the cube of b with a in as many ways as 

 possible, which is the same thing as taking the three black faces 

 with one white face, and the three white faces with one black 

 face, (3) we must take the square of a with the square of b in 

 as many ways as possible, which is the same thing as taking the 

 two black faces with the two white faces. Hence the coefficients 

 in the Binomial Theorem must be the numbers 1, 4, 6, which we 

 have already obtained in considering the cases which can arise 

 with the four counters. 



l-iG. Thus in fact Montmort argues a priori that the coeffi- 

 cients in the expansion of {a + hy must be equal to the numbers of 

 cases corresponding to the different ways in which the white and 

 black faces may appear if n counters are thrown 2)romiscuously, 

 each counter having one black face and one white face. 



Montmort gives on his page 3i a similar interpretation to 

 the coefficients of the multinomial theorem. Hence we see that 

 he in some cases passed from theorems in Chances to theorems in 

 pure Algebra, while we now pass more readily from theorems in 

 pure Algebra to their application to the doctrine of Chances. 



147. On his page 42 Montmort has the following problem: 

 There are jj dice each having the same number of faces; find the 

 number of ways in which when they are thrown at random we can 

 have a aces, b twos, c threes, . . . 



The result will be in modern notation 



\a \b[G... 



He then proceeds to a case a little more complex, namely 

 where we are to have a of one sort of faces, h of another sort, c 

 of a third sort, and so on, without specifying whether the a faces 



G— 2 



