£0$ PROBLEM IN TIIE DOCTRINE OF PERMUTATIONS* 



But, for this purpose it will be best to give determinate 

 values to m, », p, q, &c ; by which means the operation 

 will be more simple, and at the same time the law of forma- 

 tion will be equally obvious. Therefore suppose m zr 4, n 

 zz 3, p ss 2, then by actual multiplication we have 



1 + a + «* + a* + « 4 



And again, multiplying this last product by 1 + c + c% 

 we obtain the following result. 



1 + 



Ca x l C^ 1 f fl4 l 



Now, without pursuing the developement any farther, ve 

 shall readily perceive, that all the combinations in the se- 

 cond place, in both products, consist of one letter, in the 

 third place, of two letters, and in the fourth of three letters, 

 &c. And farther, that in any term, for example the fifth 

 term, the number of combinations is equal to the number 

 in the fifth, fourth, and third, of the foregoing product; the 

 number of combinations in the fourth term is equal to the 

 number in the fourth, third, and second : that is, the num- 

 ber of combinations in each term is equal to the number in 

 the three last named terms of the foregoing product; and if 

 we had used c s , then the number in each term would have 

 "been equal to the four last named terms of the foregoing 



product; and generally, if we had employed c p , the num- 

 ber 



