<p>16 persons are to be seated in a row so that in half of the seats from one side 4 particular persons sit consecutively and in the other half 3 particular persons sit together. Find the number of ways in which they can be seated.</p>
Step-by-Step Solution
Key Concept: Divide the 16 seats into two halves (8 each side), treat the 4 consecutive persons as one unit in the first half and 3 consecutive persons as one unit in the second half, then count arrangements of these units along with remaining persons.
<p><strong>Step 1:</strong> Divide 16 seats into two halves: first 8 seats (from one side) and last 8 seats (other side).</p><p><strong>Step 2:</strong> In first half (8 seats): Treat 4 particular persons as one block. We have 1 block + (8-4) = 5 units to arrange in 8 seats. The 4 persons within the block can be arranged in 4! ways. Remaining 4 persons in these 5 units can be arranged in 4! ways. Arrangements = 5! × 4! × 4!</p><p><strong>Step 3:</strong> In second half (8 seats): Treat 3 particular persons as one block. We have 1 block + (8-3) = 6 units to arrange in 8 seats. The 3 persons within the block can be arranged in 3! ways. Remaining 5 persons in these 6 units can be arranged in 5! ways. Arrangements = 6! × 3! × 5!</p><p><strong>Step 4:</strong> Total arrangements = (5! × 4! × 4!) × (6! × 3! × 5!)</p><p>= (120 × 24 × 24) × (720 × 6 × 120)</p><p>= 69,120 × 518,400</p><p>∴ Answer: <strong>5! × 4! × 4! × 6! × 3! × 5! = 35,831,808,000</strong> or <strong>5! × 6! × 4! × 3! × 4! × 5!</strong></p>
Correct Answer: 5