<p>In how many ways can a strip of cloth with 5 stripes on it be colored with 3 different colours? Assume that the strip of cloth can be flipped over i.e do not count as different patterns that are equivalent if the cloth is turned around. As example following two strips are same:</p><table border='1'><tr><td>R</td><td>W</td><td>G</td><td>R</td><td>W</td></tr><tr><td>W</td><td>R</td><td>G</td><td>W</td><td>R</td></tr></table>
Step-by-Step Solution
Key Concept: Use Burnside's lemma to count distinct colorings under the group action of reflections. The number of distinct patterns equals the average number of colorings fixed by each symmetry operation (identity and reflection).
<p><strong>Step 1:</strong> Identify the symmetry group. The cloth can be flipped over, so we have 2 symmetries: identity (I) and reflection (R) about the middle axis.</p><p><strong>Step 2:</strong> Count colorings fixed by identity. Every coloring is fixed by identity: <strong>3^5 = 243</strong> colorings.</p><p><strong>Step 3:</strong> Count colorings fixed by reflection. A coloring is fixed by reflection when stripe 1 = stripe 5, stripe 2 = stripe 4, and stripe 3 is free. This gives us <strong>3 × 3 × 3 = 27</strong> fixed colorings (independent choices for positions: (1,5), (2,4), and (3)).</p><p><strong>Step 4:</strong> Apply Burnside's lemma. Number of distinct patterns = (1/|G|) × Σ(fixed colorings) = (1/2) × (243 + 27) = (1/2) × 270 = <strong>135</strong></p><p>∴ Answer: A (135)</p>
Correct Answer: A