Arrangements and Groups. 129 



/J. In how many ways can a child be named, supposing that 

 there are 400 different Christian names, without giving it more 

 than three Christian names ? 



7^. In how many ways can vSeven people sit at a round table ? 

 i^. There are 5 straight lines in a plane, no two parallel ; 

 how many intersections are there ? 



16. On a railway there are 20 stations of a certain class. Find 

 the number of different kinds of tickets required, in order that 

 tickets may be sold at each station for each of the others. 



77. Find the number of signals that can be made with four 

 lights of different colors, which can be displayed any number at 

 a time, arranged either above one another, side by side, or 

 diagonally. 



18. From a company of 90 men, 20 are detached for mount- 

 ing guard each day ; how long will it be before the same 20 men 

 are on guard together, supposing the men to be changed as much 

 as possible ? How often will each man have been on guard during 

 this time ? 



ig. A lock contains 5 levers, each capable of being placed in 

 10 distinct positions. At a certain arrangement of the levers the 

 lock is open. How many locks of this kind can be made so that 

 no two shall have the same key ? 



20. There dre n points in a plane no three of which are in 

 the same straight line. Find the number of straight lines which 

 result from joining them. 



21. There are 71 points in a plane, no three of which are in 

 the same straight line except r, which are all in the same straight 

 line ; find the number of straight lines which result from joining 

 them. 



22. There are n points in space, no four of which are in the 

 same plane with the exception of r which are all in the same 

 plane. How many planes are there, each containing three of the 

 points ? 



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