A rectangular flag is divided into four triangles labeled left, right, top, and bottom as shown. each triangle is to be colored using one of red,
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Case 1: All four triangles have different colors:
There are 5 colors to choose for the LEFT triangle.
There are then 4 colors remaining to choose for the RIGHT triangle.
There are then 3 colors remaining to choose for the TOP triangle.
There are then 2 colors remaining to choose for the BOTTOM triangle.
That's 5*4*3*2 = 5P4 = 120 ways for Case 1.
Case 2: the LEFT triangle and the RIGHT triangle are colored alike
but the TOP and BOTTOM triangles are colored different.
There are 5 colors to choose for the LEFT and RIGHT triangles.
There are then 4 colors remaining to choose for the TOP triangle.
There are then 3 colors remaining to choose for the BOTTOM triangle.
That's 5*4*3 = 5P3 = 60 ways for Case 2.
Case 3: the TOP triangle and the BOTTOM triangle are colored alike
but the RIGHT and LEFT triangles are colored different.
That's the same number as Case 2, or 60 ways.
Total for the three cases: 120+60+60 = 240 ways to color the flag.
There are 5 colors to choose for the LEFT triangle.
There are then 4 colors remaining to choose for the RIGHT triangle.
There are then 3 colors remaining to choose for the TOP triangle.
There are then 2 colors remaining to choose for the BOTTOM triangle.
That's 5*4*3*2 = 5P4 = 120 ways for Case 1.
Case 2: the LEFT triangle and the RIGHT triangle are colored alike
but the TOP and BOTTOM triangles are colored different.
There are 5 colors to choose for the LEFT and RIGHT triangles.
There are then 4 colors remaining to choose for the TOP triangle.
There are then 3 colors remaining to choose for the BOTTOM triangle.
That's 5*4*3 = 5P3 = 60 ways for Case 2.
Case 3: the TOP triangle and the BOTTOM triangle are colored alike
but the RIGHT and LEFT triangles are colored different.
That's the same number as Case 2, or 60 ways.
Total for the three cases: 120+60+60 = 240 ways to color the flag.
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