SOLUTION: each of the six squares shown in the figure is to be filled with any one of the ten possible colors. how many ways are there of coloring the strip shown in the figure so that no tw

Question 339880: each of the six squares shown in the figure is to be filled with any one of the ten possible colors. how many ways are there of coloring the strip shown in the figure so that no two adjacent squares have the same color?
their is a purple box, a yellow box, a red box , a green box, a blue box and and an orange box lined up next to each other.
Answer by Edwin McCravy(20056) (Show Source): You can put this solution on YOUR website!

Fill the 1st square any of the 10 ways.
That's 10 ways to color the 1st square.

For every one of those 10 ways to color the 1st square, there
are 9 ways to color the 2nd square, that is, with any of the 
9 colors different from the color in the 1st square.
That's 10*9 ways to color the first 2 squares.

For every one of those 10*9 ways to color the 1st 2 squares, there
are 9 ways to color the 3rd square, that is, with any of the 
9 colors different from the color in the 2nd square.
That's 10*9*9 ways to color the first 3 squares.

For every one of those 10*9*9 ways to color the 1st 3 squares, there
are 9 ways to color the 4th square, that is, with any of the 
9 colors different from the color in the 3rd square.
That's 10*9*9*9 ways to color the first 4 squares.

For every one of those 10*9*9*9 ways to color the 1st 4 squares, there
are 9 ways to color the 5th square, that is, with any of the 
9 colors different from the color in the 4th square.
That's 10*9*9*9*9 ways to color the first 5 squares.

For every one of those 10*9*9*9*9 ways to color the 1st 5 squares, there
are 9 ways to color the 6th square, that is, with any of the 
9 colors different from the color in the 5th square.
That's 10*9*9*9*9*9 ways to color the first 6 squares. 

That's all the squares, so the answer is

10*9⁵ = 590490 ways.

Edwin

RELATED QUESTIONS
The problem is about color arrangements . I solved it but don't know if the answer is... (answered by stanbon)

Ralph has seven different colors of leftover paint. Since there is not enough of any one... (answered by stanbon)

There are 12 color pencils in different colors. If the 4 boxes are filled with different... (answered by ewatrrr,ikleyn)

There are 12 color pencils in different colors. If the 4 boxes are filled with different... (answered by ikleyn)

There are 5 colors pens in red, yellow, blue, green, and purple. How many patterns can be (answered by Boreal,ikleyn)

A. There are 5 colors pens in red, yellow, blue, green, and purple. How many patterns can (answered by ikleyn)

SEVEN HOUSES IN A ROW ARE TO BE PAINTED WITH ONE OF THE COLORS RED, BLUE, GREEN OR... (answered by sudhanshu_kmr)

A puzzle requires you to find a corredt three color code. There are only three colors... (answered by jorel1380)

There are six available colors. find the number of ways to color the unit cells of a 2 x... (answered by ikleyn)