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# Probability with Casework

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Each of eight houses on a street is painted brown, yellow or white. Each house is painted only one color and each color is used on at least one house. No two colors are used to paint the same number of houses. In how many ways could the eight houses on the street be painted?

Apr 27, 2018

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Each of eight houses on a street is painted brown, yellow or white. Each house is painted only one color and each color is used on at least one house. No two colors are used to paint the same number of houses. In how many ways could the eight houses on the street be painted?

Well this is what I am thinking

Let the colours be A B and C

There are 3!=6 possible variations on which colour goes with which letter

Now the house numbers can be  5,2 and 1    OR   4,3,1

There can be no other combination.

so we have

$$6(\frac{8!}{5!2!}+\frac{8!}{4!3!})=6(168+280)$$

=2688 possible combination.

Apr 27, 2018