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/
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