<p>The number of ways of distributing 8 identical balls in 3 distinct boxes so that none of the boxes is empty is</p>
Step-by-Step Solution
Key Concept: Use stars and bars formula with the constraint that each box must have at least one ball. First place one ball in each box, then distribute remaining balls freely.
<p><strong>Step 1:</strong> Identify the constraint - each of 3 distinct boxes must contain at least one ball from 8 identical balls.</p><p><strong>Step 2:</strong> Since none of the boxes can be empty, place 1 ball in each box first. This uses 3 balls, leaving 8 - 3 = 5 identical balls to distribute.</p><p><strong>Step 3:</strong> Now distribute the remaining 5 identical balls into 3 distinct boxes with no restrictions. Using stars and bars formula: C(n+k-1, k-1) where n = 5 (remaining balls) and k = 3 (boxes).</p><p><strong>Step 4:</strong> Number of ways = C(5+3-1, 3-1) = C(7, 2) = (7 × 6)/(2 × 1) = 21</p><p><strong>Verification:</strong> This can also be expressed as the coefficient of x⁸ in (x + x² + x³ + ...)³ = x³(1 + x + x² + ...)³ = x³/(1-x)³, which gives C(8-1, 3-1) = C(7, 2) = 21</p><p>∴ Answer: <strong>21</strong> (Option D)</p>
Correct Answer: D