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Zorluk: OrtaBasic Probability and Counting Methods

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?

Cevap: 80 flag designs

Cevap

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.

Adım Adım Çözüm

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.

Anahtar Kavram

Fundamental Counting Principle with adjacent restrictions
Tahmini Süre:1m 15s
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