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The Three Little Pigs Arrange Their Houses

Date: 05/13/2002 at 13:33:38
From: D. Boiros
Subject: Word Problem

My daughter got a problem work sheet on the Three Little Pigs and 
one of the questions we came up with two different answers for. 

The question was: How many different ways could the pigs arrange 
their houses?   

I come up with 27 by using 3*3*3 = 27.  But when we go to list 
the different combinations we can only get 6.  Could you please 
tell us the correct answers? 

Thank you.
Mrs. Boiros


Date: 05/13/2002 at 16:13:23
From: Doctor Jeremiah
Subject: Re: Word Problem

Hi there Mrs. Boiros!  I am glad you wrote.

I think this is a very interesting question. The answer you 
computed is too large because it includes arrangements that 
include the same house more than once. 

Let A, B, and C represent the houses belonging to the first, 
second, and third pigs respectively. 

All the possible arrangements are:

 A A A  A A B  A A C  A B A  A B B  A B C  A C A  A C B  A C C
 B A A  B A B  B A C  B B A  B B B  B B C  B C A  B C B  B C C   
 C A A  C A B  C A C  C B A  C B B  C B C  C C A  C C B  C C C

But notice that this includes using the same house multiple times 
(which obviously really isn't possible with houses).

The reason it comes out to 27 is because the first house can be 
one of 3 choices, and the second house can be one of 3 choices, 
and the third house can also be one of 3 choices, so the number 
of possible arrangements is:

  3 (first) x 3 (second) x 3 (third) = 27

Now, like we said before, this isn't the right way to go for 
houses because you can't use the same house multiple times, so 
the first house is a choice of 3, and then the second house
in the row is a choice of 2 (because one is already used), and 
then the third house isn't a choice at all (because the other two 
are already used).  

So the total number of arrangements (excluding duplicates) would 
be: 

  3 (first) x 2 (second) x 1 (third) = 6

Now we have a lot fewer possible arrangements!

  - - -  - - -  - - -  - - -  - - -  A B C  - - -  A C B  - - -
  - - -  - - -  B A C  - - -  - - -  - - -  B C A  - - -  - - -
  - - -  C A B  - - -  C B A  - - -  - - -  - - -  - - -  - - -

Does that help explain the difference?

Here is a bit more info on this and more complicated topics:

  http://www.mathforum.org/dr.math/faq/faq.comb.perm.html 

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

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