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Permutations of Vertices


Date: 04/22/97 at 20:39:36
From: Phil Steen
Subject: Hard math problem

There is this hard question that I would like you to help me with:

A triangle ABC can be named in 6 different ways using its vertices: 
ABC, ACB, BCA, BAC, CAB, and CBA.	

A. In how many different ways can a quadrilateral be named?
B. In how many different ways can a pentagon be named?


Date: 05/21/97 at 20:51:54
From: Doctor Sydney
Subject: Re: Hard math problem

Dear Phil,

This is a good question!  I bet if we think carefully about how you 
found your answer for the triangle, you will be able to come up with 
the answers to your two questions.  

Let's re-examine how you discovered that there are 6 ways to name 
the triangle ABC.  It looks as if you began with the vertex A and said 
to yourself,  "Hmmm, how many different ways can I name this triangle 
with the first letter is A?  Well, I can start with A and then 
continue labeling in a clockwise direction, or I can move in a 
counterclockwise direction. Going in one direction gives me ABC and 
going in the other direction gives me ACB. So, there are two ways to 
name this triangle where the first letter in the name is 'A.'" 

Then what did you do? Well, perhaps you thought to yourself, "Okay, 
now I know that there are two names that begin with the letter A, 
but there are still 2 letters left that the name might start with: 
B and C.  So, I still need to figure out how many ways I can name the 
triangle such that it starts with a B or a C."  

Wanting to continue the problem in an orderly fashion, you can proceed 
in alphabetical order to think about how many ways there are to name 
the triangle that starts with the letter B.  Just as in the case with 
names that start with A, you can continue naming the triangle by going 
in the clockwise direction from B or the counterclockwise direction 
from B, so there are two names you can give to the triangle that 
begins with the letter B.

At this point, you are very close to solving the problem - you just 
have to figure out how many ways you can name the triangle such that 
C is the first letter in the name.  There are two ways to name it: you 
can move clockwise from C or counterclockwise from C.

Since there are two ways to name the triangle that begins with A, two 
ways for the triangle that begins with B, and two ways for the 
triangle that begins with C, there is a total of 6 ways to name the 
triangle.

Now, how can you apply the above method to solving your two problems?  
Use the same thought process. First choose a vertex/letter and figure 
out how many ways you can name the triangle where that letter is the 
first letter in the name. Do this for each vertex/letter in the 
figure. Then add these up to get the final answer. You can go through 
this same process to find answers to your problems.

If any of this is confusing, or you need more help, please do write 
back.  We'd be happy to help out!  Good luck!

-Doctor Sydney,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Permutations and Combinations

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