Question:

<p><strong>Step 1: Identify the distribution</strong></p><p>We need to distribute 22 books to 5 students where: 2 students get 5 books each, and 3 students get 4 books each. Check: 2(5) + 3(4) = 10 + 12 = 22 ✓</p><p><strong>Step 2: Choose which 2 students get 5 books</strong></p><p>Select 2 students from 5 to receive 5 books: C(5,2) = 10 ways</p><p><strong>Step 3: Partition books into the required groups</strong></p><p>Distribute 22 different books into groups of sizes 5, 5, 4, 4, 4 and assign them to the students:</p><p>• Choose 5 books for first selected student: C(22,5)</p><p>• Choose 5 books for second selected student: C(17,5)</p><p>• Choose 4 books for third student: C(12,4)</p><p>• Choose 4 books for fourth student: C(8,4)</p><p>• Remaining 4 books go to fifth student: C(4,4) = 1</p><p><strong>Step 4: Calculate total</strong></p><p>Total ways = C(5,2) × C(22,5) × C(17,5) × C(12,4) × C(8,4) × C(4,4)</p><p>= 10 × (26334) × (6188) × (495) × (70) × (1)</p><p>= 10 × C(22,5) × C(17,5) × C(12,4) × C(8,4)</p><p><strong>Calculation:</strong></p><p>C(22,5) = 26,334</p><p>C(17,5) = 6,188</p><p>C(12,4) = 495</p><p>C(8,4) = 70</p><p>= 10 × 26,334 × 6,188 × 495 × 70 = <strong>568,032,638,400</strong></p><p>Or expressed as: <strong>22!/(5!×5!×4!×4!×4!) × 5!/(2!×3!) = (22!)/(2!×3!) ÷ (5!×5!×4!×4!×4!)</strong></p><p>∴ Answer: <strong>568,032,638,400</strong> or <strong>22!/(2!×3!) ÷ (5!×5!×4!×4!×4!)</strong></p>