Problem: Seven houses in a row are to be painted with one of the colors red, blue, green or yellow. In how many different ways can the houses be painted so that no two adjacent houses are the same color? My answer is 2916 but I am really unsure of it. I don't understand the formula that should be used to find this answer. Thanks.
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probability
- Thread starter luvbug
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luvbug said:Problem: Seven houses in a row are to be painted with one of the colors red, blue, green or yellow. In how many different ways can the houses be painted so that no two adjacent houses are the same color? My answer is 2916 but I am really unsure of it. I don't understand the formula that should be used to find this answer. Thanks.
Why are you having doubts about your answer? - what tells you this is incorrect?
How did you arrive at the answer?
Hello, luvbug!
We can baby-talk our way through this one . . .
\(\displaystyle \begin{array}{cc}\text{The 1st house can be any of the four colors: }\;\; & \text{4 choices} \\ \text{The 2nd house must not match the 1st house:} & \text{3 choices} \\ \text{The 3rd house must not match the 2nd house:} & \text{3 choices} \\ \text{The 4th house must not match the 3rd house:} & \text{3 choices} \\ \text{The 5th house must not match the 4th house:} & \text{3 choices} \\ \text{The 6th house must not match the 5th house:} & \text{3 choices} \\ \text{The 7th house must not match the 6th house:} & \text{3 choices} \end{array}\)
\(\displaystyle \text{Therefore, there are: }\:4\cdot3^6 \;=\;2916\text{ ways.}\)
We can baby-talk our way through this one . . .
Seven houses in a row are to be painted with one of the colors: red, blue, green or yellow.
In how many different ways can the houses be painted so that no two adjacent houses are the same color?
My answer is 2916. . Correct! How did you get it?
\(\displaystyle \begin{array}{cc}\text{The 1st house can be any of the four colors: }\;\; & \text{4 choices} \\ \text{The 2nd house must not match the 1st house:} & \text{3 choices} \\ \text{The 3rd house must not match the 2nd house:} & \text{3 choices} \\ \text{The 4th house must not match the 3rd house:} & \text{3 choices} \\ \text{The 5th house must not match the 4th house:} & \text{3 choices} \\ \text{The 6th house must not match the 5th house:} & \text{3 choices} \\ \text{The 7th house must not match the 6th house:} & \text{3 choices} \end{array}\)
\(\displaystyle \text{Therefore, there are: }\:4\cdot3^6 \;=\;2916\text{ ways.}\)