Permutations & Combinations
Circular Arrangements / Garland Problems
Grade 11
Question:
<p>Find the number of ways in which 8 different flowers can be strung to form a garland so that four particular flowers are never separated.</p>
Step-by-Step Solution
Key Concept: Treat the 4 particular flowers as a single unit, then arrange this unit with the remaining 4 flowers. Account for circular arrangements and internal arrangements of the locked group.
<p><strong>Step 1:</strong> Treat 4 particular flowers as one single unit/block.</p><p><strong>Step 2:</strong> Now we have 5 objects to arrange: [Block of 4] + 4 other flowers = 5 objects in a circle.</p><p><strong>Step 3:</strong> Circular arrangements of 5 objects = (5-1)! = 4! = 24</p><p><strong>Step 4:</strong> The 4 flowers within the block can be arranged among themselves in 4! = 24 ways.</p><p><strong>Step 5:</strong> Total arrangements = 24 × 24 = 576</p><p><strong>Step 6:</strong> Since a garland can be flipped over (both directions look identical), divide by 2: 576 ÷ 2 = 288</p><p>∴ Answer: <strong>288</strong></p>
Correct Answer: 288