PERMUTATION AND COMBINATION. 229 



2. Take the series i,. 2, 3, 4, &c. up to the number of 

 given things of the first sort, and the series i, 2, 3, 4, 

 &c. up to the number of given things of the second 

 sort, &c. 



3. Divide the product of all the terms of the first se- 

 ries by the joint product of all the terms of the remaining 

 ones, and the quotient will be the answer required. 



EXAMPLES. 



I. How many variations may be made of the letters in 

 the word Bacchanalia ? 



I X 2(=:number of rs)=: 2 

 I X 2X3=4( = number of as) = 24 

 iX2X3X4X5X<5X7XSX9XioXii(= number of let- 

 ters in the word) = 399 1 6800 



2X24 = 48)39916800(831600 the answer. 

 151 

 76 

 288 



How 



one ; which is = .^ ± — 1 = 1. Again, if three of die quan- 



1X2x3X4 

 tities only be the same, or ahcd become aaah, the number of va_ 

 riations will be reduced to these 4, aaahy aabay ahaa and haaa ; 



which is =: —^ T =4.. And thus it may be shewn, that, 



1X2x3 



if two of the quantities be alike, or the 4 quantities be aahc, the 



number of variations will be reduced to 12 ; which may be ex- 



pressed by i2<i><32l4 ^ ,,. 

 1X2 



And by reasoning in the same manner, it will appear, that the 



number of changes, which can be made of the quantities alhccc, is 



equal to 60 ; which may be expressed by ^X^X3X 4X5x6 



1X2x1X2x3 

 ;=6o ; and so of any other Quantities whatever. 



