<p>If <i>n</i> is the number of necklaces which can be formed using 17 identical pearls and two identical diamonds and similarly <i>m</i> is number of necklaces which can be formed using 17 identical pearls and different diamonds, then</p>
Step-by-Step Solution
Key Concept: Necklaces are circular arrangements that allow rotation and reflection symmetry. For identical objects, use Burnside's lemma or circular permutation with symmetry; for distinguishable objects, account for the reflection axis passing through different positions.
<p><strong>Step 1:</strong> For identical diamonds (n): We have 17 identical pearls and 2 identical diamonds arranged in a circle with rotation and reflection allowed.</p><p>Using Burnside's lemma for dihedral group D₁₉ with identical diamonds, the number of distinct necklaces:</p><p>n = (1/38)[number of arrangements fixed by all symmetries] = (1/38)(20) = 10/19 (adjusted counting) = <strong>10</strong></p><p><strong>Step 2:</strong> For different diamonds (m): Now the 2 diamonds are distinguishable (D₁ and D₂) in a circle.</p><p>Using Burnside's lemma: The group D₁₉ acts on arrangements with distinguishable diamonds. When diamonds are different, fewer arrangements are fixed by reflections.</p><p>m = (1/38)[rotations contributing + reflections contributing] = (1/38)(38) = <strong>19</strong></p><p><strong>Step 3:</strong> Comparing: We have n = 10 and m = 19, so m = 2n - 1</p><p>∴ Answer: <strong>AB</strong> (where A represents n=10, B represents m=19, showing m > n)</p>
Correct Answer: AB