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Color for first strip can be selected from 4 colors in 4 ways.

Color for next strip can be selected in 3 ways(since we can't allow adjacent strips to have same color)

For similar reason, color for each of the next 3 strips can be selected in 3 ways .

Hence total number of ways of coloring the strips of the flag = $4*3*3*3*3$ = $324$ ways

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