<p>Twelve different letters are to be put in twelve pockets in a row. If five of the pockets are too small for six of the letters then in how many different ways can the letters be put in the pockets?</p>
Step-by-Step Solution
Key Concept: Identify that 6 letters cannot go into 5 small pockets, so these 6 letters must go into the 7 remaining large pockets. The remaining 6 letters can go anywhere, but we must ensure the constraint is satisfied.
<p><strong>Step 1:</strong> Identify the constraint. We have 12 pockets: 5 small and 7 large. We have 12 letters: 6 that are too big for small pockets and 6 that fit anywhere.</p><p><strong>Step 2:</strong> The 6 large letters MUST go into the 7 large pockets. Choose 6 positions from 7 large pockets: C(7,6) = 7 ways. Arrange 6 large letters in these positions: 6! ways.</p><p><strong>Step 3:</strong> The remaining 1 large pocket and 5 small pockets (total 6 pockets) will hold the 6 small letters. Arrange 6 small letters in these 6 remaining pockets: 6! ways.</p><p><strong>Step 4:</strong> Total arrangements = C(7,6) × 6! × 6! = 7 × 720 × 720 = 7 × 518400 = 3,628,800</p><p>Alternatively: Choose which 6 of the 7 large pockets get the large letters (7 ways), arrange large letters (6! ways), arrange small letters in remaining 6 pockets (6! ways).</p><p>∴ Answer: <strong>7 × 6! × 6! = 3,628,800</strong></p>
Correct Answer: 7