226 ARITHMETIC. 



PROBLEM II. 



\Afiy nuniher of different things being given y to find honv rtian^ 

 changes c^n be made out of thcm^ by taking any given num* 

 her at a ti?ne» 



RULE* 



Take a series of numbers, beginnlnp; at the number of 

 things given, and decreasing by i till the number o^, terms 

 be equal to the number of things to be taken at a time, 

 and the product of all the term.s will be the answer re- 

 quired. 



EXAMPLES. 



* This rule, expressed in terms, is as follows : myim — i X f^i — 2 

 Xm — 3, &c. to n terras ; where m = number of things given, 

 and n = quantities to be taken at a time. 



In order to demonstrate the rule, It will be necessary to pre- 

 mise the following 



.L E M M A. 



The number of change^ of m things, taken ri at a time, is equal 

 to m changes of m — i things, taken n — -i at a time. • 



Demonstration. Let any 5 quantities, abcde^ be given. 



First, leave out the a, and let ir -^ number of all the variations 

 of every two, be. Id, &c. that can be taken out of the 4 remain- 

 ing quantities, bcde. 



Now let ^ be put in the first place of each of them, abc^ ahd^ 

 Sec. and the nuMber of changes will' st?ll remain the same ; that is» 

 •0 =: number of variations of every 3 out of the 5, abcde, when a 

 13 first. 



In like manner, If b, c, d, e, be successively left out, the "num- 

 ber of variations of all the twos will also = v ; and b, r, d, e^ 

 being respectively put in the first place, to make 3 quantities out 

 ©f 5, there will still be tf variations as before. 



But these are all the variations, that can happen of 3 things out 

 «f 5, when a, by c, d^e, are successively |^ut first ; and therefore 



the 



