<p>If 12 identical balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is</p>
<p>\(\dfrac{55}{3}\left(\dfrac{2}{3}\right)^{11}\)</p>
<p>\(55\left(\dfrac{2}{3}\right)^{10}\)</p>
<p>\(220\left(\dfrac{1}{3}\right)^{12}\)</p>
<p>\(22\left(\dfrac{1}{3}\right)^{11}\)</p>
Step-by-Step Solution
Key Concept: Since boxes are identical, we need to count partitions of 12 into 3 parts (unordered). The probability equals (number of partitions with exactly one part = 3) divided by (total partitions of 12 into 3 parts).
<p><strong>Step 1:</strong> Find all partitions of 12 into exactly 3 positive parts (since boxes are identical, order doesn't matter).</p><p>The partitions are: (1,1,10), (1,2,9), (1,3,8), (1,4,7), (1,5,6), (2,2,8), (2,3,7), (2,4,6), (2,5,5), (3,3,6), (3,4,5), (4,4,4)</p><p><strong>Step 2:</strong> Count total partitions: 12 distinct partitions exist.</p><p><strong>Step 3:</strong> Count favorable partitions where exactly one box has 3 balls: (1,3,8), (2,3,7), (3,3,6), (3,4,5) → 4 partitions.</p><p><strong>Step 4:</strong> Calculate probability = 4/12 = 1/3.</p><p>∴ Answer: D</p>
Correct Answer: D