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Introduction to Permutations by Standing in Line

Date: 11/18/2004 at 16:02:37
From: Robert
Subject: Math

Hi!  I think I need help with this question.  Six persons will stand
in line.  In how many different ways can they stand in line?

I don't understand the question.  Could you help me?



Date: 11/19/2004 at 13:55:00
From: Doctor Edwin
Subject: Re: Math

Hi, Robert.

We could try to answer this question by thinking about a line of 6 
people, or we could try to answer this question by thinking about a 
line of 1 person and working our way up.  But I think it will be fun 
to start in the middle, with a line of 3 people.

Let's say my friends A, B, and C are in a line.  A wants to be first. 
How many ways could they all line up if they let A be first?  It's 
two, right?  They can be ABC or ACB.  That is, the number of possible 
orderings with A first is the same as the number of orderings of B 
and C.  So the number of orderings of n people if you know which one 
is first is the number of orderings of n-1 people. 

Of course, A didn't HAVE to be first, did he?  B could have been 
first, and C could have been first. So there are two orderings where A 
is first, two where B is first, and two where C is first.  Which means 
that the number of orderings for 3 people is 3 times the number of 
orderings for two people.

Now we know that the number of orderings for three people is 6.  Oh, 
oh, here comes my friend D, and he wants to play, too!  A, B, and C 
are nice, so they let D be first in line, so that means there are 6 
orderings of my 4 friends when D is first.  But of course A, B, or C 
could also have been first.  So there are 6 orderings with D first, 6 
with A first, 6 with B first, and 6 with C first.

In other words, the number of orderings of 4 people is 4 * (the 
number of orderings of 3 people).  But we know the number of orderings 
of 3 people is 3 * (the number of orderings of 2 people).  That means 
that the number of orderings of 4 people is 4 ( 3 * (the number of 
orderings of 2 people)). 

Finally, how many orderings are there of 1 person?  Well, one, right? 
So the number of orderings of 2 people is 1 with A in front and 1 
with B in front, or 2 * (the number of orderings of 1 person).  So 
that means that the number of orderings of 4 people is:

 4 * (3 * (2 * 1)))

Of course, we don't need the parentheses.  So we get

 4 * 3 * 2 * 1

Now, 5 people!  E walks up.  He's not real popular, but they let him 
play anyway.  Each one takes a turn being first.  When he does, there 
are 4 * 3 * 2 * 1 orderings.  Each of the 5 friends takes a turn being 
first, meaning there are 5 times that many orderings.  You see where 
this is going?

So, how many ways can you order 6 people?

- Doctor Edwin, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Permutations and Combinations

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