230 ARITHMETIC. 



2. Hovy many different numbers can be made of the 

 fciiowing figures, 1220005555 ? Ans. 12600. 



3. What is the variety in the succession of the follow- 

 ing musical notes, fa, fa, fa, sol, sol, la, mi, fa ? 



Ans. 3360. 



PROBLEM IV. 



fTo find the changes of any g'rjen number of things, tahen a 

 given number at a time ; in *which there are several given 

 things of one sorty several of another , ^c» 



RULE.* 



1. Find all the different forms of combination of all the 

 given things, taken as many at a time as in the question. 



2. Find the number of changes in any form, and mul- 

 tiply it by the number of combinations in that form. 



3. Do the same for every distinct form ; and the sum 

 of all the products will give the- whole number of changes 

 required. 



Note. To find the different forms of combination proceed 

 thus : 



1. Place the things so, that the greatest indices may be 

 first, and the rest in order. 



2. Begin with the first letter, and join it to the second, 

 third, fourth, &c. to the last. 



3. Then take the second letter, and join it to the third, 

 fourth, &c. to the last ; and so on through the whole, 

 always remembering to reject such combinations as have 

 occurred before j and this will give the combinations of 

 all the tv/os. 



4. Join 



* The reason of this rule is plain from what has been shewn 

 before, and the nature of the problem. 



