128 Algebra. 



Since by Art. 8, groups where repetitions are allowed can be 

 expressed in terms of groups when repetitions are not allowed, it 

 would be an easy matter to obtain equations involving groups 

 with repetitions. 



13. Examples and Problems. 



1. How many different groups of two each can be made 

 from the letters a, d, /, n, sf See VIII, Art. 5. 



2. How many arrangements of five each can be made from 

 the letters of the word G^-oups f 



J. How many different signals can be made with five flags 

 of different colors hoisted one above another all at a time ? 



4. How many different signals can be made from seven 

 flags of different colors hoisted one above another, five at a time ? 



5. How many different groups of 1 3 each can be made out 

 of 52 cards, no two alike? 



6. How many different signals can be made from five flags 

 of different colors, which can be hoisted any number at a time 

 above one another ? 



7. How many different signals can be made from seven flags 

 of which 2 are red, i white, 3 blue, i yellow when all are dis- 

 played together, one above another, for each signal. 



8. A certain lock opens for some arrangement of the num- 

 bers o, I, 2, 3, 4, 5, 6, 7, 8, 9, taken 6 at a time, repetitions 

 allowed. How many trials must be made before we would be 

 sure of opening the lock ? 



p. In how many ways can a committee of 3 appointed from 

 5 Germans, 3 Frenchmen and 7 Americans, so that each nation- 

 ality is represented ? 



10. How many different arrangements can be made of nine 

 ball players, supposing only two of them can catch and one pitch ? 



11. How many different products of three each can be made 

 from the four letters a, b, c, df 



12. In how many different ways can the letters of the word 

 algebra be written, using all the letters ? 



