<p>Fifteen identical balls have to be put in five different boxes. Each box can contain any number of balls. The total number of ways of putting the balls into the boxes so that each box contains at least two balls is equal to</p>
Step-by-Step Solution
Key Concept: Convert the constraint 'at least 2 balls per box' into an unrestricted problem by first distributing 2 balls to each of the 5 boxes (using 10 balls), then distributing the remaining 5 identical balls freely among 5 different boxes using stars and bars.
<p><strong>Step 1: Handle the constraint</strong></p><p>Each box must contain at least 2 balls. First, place 2 balls in each of the 5 boxes, using 2×5 = 10 balls.</p><p><strong>Step 2: Distribute remaining balls</strong></p><p>Remaining balls = 15 - 10 = 5 balls. Now we need to distribute 5 identical balls into 5 different boxes with no restrictions.</p><p><strong>Step 3: Apply stars and bars</strong></p><p>Using the stars and bars formula, the number of ways to distribute n identical objects into k different boxes is C(n+k-1, k-1).</p><p>Here: n = 5, k = 5</p><p>Number of ways = C(5+5-1, 5-1) = C(9, 4) = 9!/(4!×5!) = (9×8×7×6)/(4×3×2×1) = 3024/24 = 126</p><p>∴ Answer: <strong>126</strong></p>
Correct Answer: A