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Basic Probability and Counting Methods

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A student is creating a flag design consisting of 3 vertical stripes in a row. The student can choose from 5 different colors: Red, Blue, Green, Yellow, and White. To ensure visual contrast, any 2 adjacent stripes must be different colors. How many different flag designs can be created under these rules?

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Answer: 80

Answer

80
To find the total number of flag designs, apply the Fundamental Counting Principle across the three sequential stripe choices. The first stripe can be any of the 5 colors. The second stripe cannot match the first stripe's color, leaving 4 options. The third stripe cannot match the second stripe's color, which also leaves 4 options (since it can re-use the first stripe's color). Multiplying these choices together gives 5×4×4=805 \times 4 \times 4 = 80 unique flag designs.

Step-by-Step Solution

1
Determine choices for the first stripe
5 choices
Any of the 5 available colors can be selected for the first stripe.
2
Determine choices for the second stripe
4 choices
The second stripe must differ in color from the first stripe.
3
Determine choices for the third stripe
4 choices
The third stripe must differ in color from the second stripe (it may be the same color as the first stripe).
4
Apply the Fundamental Counting Principle
5×4×4=805 \times 4 \times 4 = 80
Multiply the options available at each stage to calculate the total number of distinct flag designs.

Key Concept

Fundamental Counting Principle with adjacent restrictions
Estimated Time:1m 15s
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