PERMUTATION AND COMBINATION. 2^7 



EXAMPLES. 



1. How many changes may be made out of the 3 letters 

 abcy by taking 2 at a time 



3 



2 Or 3X2e:6 the answer. 



6 The changes. 



ab 

 ha 

 ac 

 ca 

 he 

 ch 



2. How many words can be made with 5 letters of the 

 alphabet, it being admitted that a number of consonants 

 may make a word ? Ans. 5100480. 



PROBLEM 



tlje sum of all these is the sum of all the changes of 3 things out 



ofS- . 



But the sum of these is so many times ^', as is the number of 



things ; that is, 51;, or wv, =; all the changes of 3 things out of 

 5, And the same way of reasoning may be applied to any num- 

 bers whatever. 



Demonstration of the Rule. Let any 7 things, alcdef^j 

 be given, and let 3 be the number of quantities to be taken. 



Then m=7, and «=3. 



Now it is evident, that the number of changes, that can be 

 made by taking l by i out of 5 things, will be 5, which let rr-u. 



Then, by the lemma, when mz=:() and «=2, the number of 

 changes will =:m'y=6X5 ; which let =:i? a second time. 



Again by lemma,, when w=:7 and n=:3, the number of changes 



=;nTJ=:7X6X5 ; that is, mvz=.m%m — i Xw — 2, continued to 

 3, or n terms. And the same may be shev/n for any other 

 numbers. 



